SKKU_June_2017.sagews

Soliton Cellular Automata Using Rigged Configurations

Travis Scrimshaw

This is joint work with Xuan Liu (UMN).

The classical box-ball system of Takahashi and Satsuma is a nonlinear discrete dynamical system where

• states are an empty box (vacuum) or a ball;
• time evolution is given by moving balls from right to left and putting them in the next available position.

Below, we represent vacuums by $1$ and balls by $2$.

BBS = SolitonCellularAutomata('1211221211222', ['A',1,1])
l = BBS.latex_states_word(5)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 2 2 2 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

BBS = SolitonCellularAutomata('112111221111122211112222', ['A',1,1])
l = BBS.latex_states_word(10)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

BBS = SolitonCellularAutomata('11111211111222112211111112222', ['A',1,1])
l = BBS.latex_states_word(15)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 10 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 11 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 12 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 13 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 14 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

The box-ball system has solitonic behavior:

• Collections of adjacent balls, called solitons, move with speed according to their length/size.
• After interaction, called scattering, the solitons retain their sizes.

This terminology comes from the fact that the box-ball system is an ultradiscrete version of the Kortewig-de Vries (KdV) nonlinear PDE. The KdV equation models water in a thin channel (which we can take as 1-dimensional) and has solutions of solitary waves called that propograte proporitional to their speed and separate after interaction. Below, we give a discrete version of the KdV equation due to Hirota, but it still exhits the same behavior as the KdV equation solutions.

 D = DiscreteKdV(1, 5, [5,2,1,1,2,7,1,1,3,4] + [5]*40 + [5,5,5,2,2,2,3,4,5,5,5,5])
D.animation(500, step=5)

We can generalize the box-ball system to multicolored balls, where we first do time evolution on the largest color first, then down to the smallest color. This still exhibits solitonic behavior, where the solitons are weakly increasing words from left to right.

BBS = SolitonCellularAutomata('114111251111124411112335', ['A',4,1])
l = BBS.latex_states_word(10)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 3 5 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} 2 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} 2 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} 2 3 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 2 5 {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} 2 3 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 5 2 {\color{gray} 1} 2 4 4 {\color{gray} 1} 3 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 5 {\color{gray} 1} 2 4 {\color{gray} 1} 4 {\color{gray} 1} 3 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} 3 4 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} 3 5 {\color{gray} 1} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots 2 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

BBS = SolitonCellularAutomata('114111521111124411112335', ['A',4,1])
l = BBS.latex_states_word(10)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 5 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 3 5 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 5 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 5 2 {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} 2 3 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} 2 2 5 4 {\color{gray} 1} 4 {\color{gray} 1} 2 3 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 2 5 {\color{gray} 1} {\color{gray} 1} 4 3 4 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 5 {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 4 4 5 3 {\color{gray} 1} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} 5 {\color{gray} 1} {\color{gray} 1} 3 {\color{gray} 1} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 5 {\color{gray} 1} {\color{gray} 1} 3 {\color{gray} 1} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 5 {\color{gray} 1} {\color{gray} 1} 3 {\color{gray} 1} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots 2 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 5 {\color{gray} 1} {\color{gray} 1} 3 {\color{gray} 1} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

Notice that in the last example that $52$ is not a soliton as it travels with speed 1 and is separated upon scattering.

BBS = SolitonCellularAutomata('1141113311221112451112344', ['A',4,1])
l = BBS.latex_states_word(14)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 4 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} 3 3 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} 2 4 5 {\color{gray} 1} {\color{gray} 1} 2 3 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} 3 3 {\color{gray} 1} 2 2 2 {\color{gray} 1} {\color{gray} 1} 4 5 {\color{gray} 1} 2 3 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 3 3 2 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 5 3 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} 3 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 4 5 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 2 {\color{gray} 1} {\color{gray} 1} 4 4 5 {\color{gray} 1} 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 {\color{gray} 1} 2 {\color{gray} 1} 4 5 {\color{gray} 1} {\color{gray} 1} 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} 5 {\color{gray} 1} {\color{gray} 1} 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} 5 {\color{gray} 1} 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 5 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 10 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 11 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 12 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 13 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

We can generalize the box-ball system even further by using a certain class of crystals in affine type called Kirillov–Reshetikhin (KR) crystals. The idea comes from row-to-row transfer matrices of solvable lattice models. We represent elements of the box-ball system by the type $A_n^{(1)}$ (or $\widehat{\mathfrak{sl}}_{n+1}$) KR crystals $B^{1,1}$: the set $\{1, \dotsc, n+1\}$ with crystal operators $f_i(b) = \begin{cases} b + 1 \mod{n+1} & \text{if } i = b \mod{n+1}, \\ 0 & \text{otherwise}, \end{cases} \qquad e_i(b) = \begin{cases} b - 1 \mod{n+1} & \text{if } i = b+1 \mod{n+1}, \\ 0 & \text{otherwise}. \end{cases}$ Note that the vacuum state corresponds to the element $1$.

We also need the KR crystal $B^{1,s}$, which are monomials in $B^{1,1}$ considered as commuting variables and the crystal operators act as above. Crystals also have a natural tensor product structure. Moreover, there is a unique crystal isomorphism $R \colon B^{1,s} \otimes B^{1,s'} \to B^{1,s'} \otimes B^{1,s},$ called the combinatorial $R$-matrix, that is given by $b \otimes b' \mapsto \widetilde{b}' \otimes \widetilde{b}$ if and only if $b \xleftarrow{\operatorname{RSK}} b' = \widetilde{b}' \xleftarrow{\operatorname{RSK}} \widetilde{b}.$ Last, we require the local energy function, $H \colon B^{1,s} \otimes B^{1,s'}$ defined as $H(b \otimes b')$ as the maximal number of boxes the row $b'$ can cover of $b$.

For more background on type $A_n^{(1)}$ crystals, see Crystals for dummies by Mark Shimozono.

K = crystals.KirillovReshetikhin(['A',3,1], 1, 2)
latex.eval(latex(K.digraph()))

''

Kp = crystals.KirillovReshetikhin(['A',3,1], 1, 3)
R = K.R_matrix(Kp)
H = K.local_energy_function(Kp)
for x in R.domain()[140:160]:
    ascii_art(x, '   |--->   ', R(x), '       H = ', H(x))
    print("")

2 4 # 1 1 1 |---> 1 2 4 # 1 1 H = 2 2 4 # 1 1 2 |---> 1 2 4 # 1 2 H = 2 2 4 # 1 2 2 |---> 2 2 4 # 1 2 H = 2 2 4 # 2 2 2 |---> 2 2 4 # 2 2 H = 1 2 4 # 1 1 3 |---> 1 2 4 # 1 3 H = 2 2 4 # 1 2 3 |---> 2 2 4 # 1 3 H = 2 2 4 # 2 2 3 |---> 2 2 4 # 2 3 H = 1 2 4 # 1 3 3 |---> 2 3 4 # 1 3 H = 2 2 4 # 2 3 3 |---> 2 2 4 # 3 3 H = 1 2 4 # 3 3 3 |---> 2 3 4 # 3 3 H = 1 2 4 # 1 1 4 |---> 2 4 4 # 1 1 H = 2 2 4 # 1 2 4 |---> 2 4 4 # 1 2 H = 2 2 4 # 2 2 4 |---> 2 2 4 # 2 4 H = 1 2 4 # 1 3 4 |---> 2 4 4 # 1 3 H = 2 2 4 # 2 3 4 |---> 2 2 4 # 3 4 H = 1 2 4 # 3 3 4 |---> 2 3 4 # 3 4 H = 1 2 4 # 1 4 4 |---> 2 4 4 # 1 4 H = 1 2 4 # 2 4 4 |---> 2 4 4 # 2 4 H = 1 2 4 # 3 4 4 |---> 2 4 4 # 3 4 H = 1 2 4 # 4 4 4 |---> 2 4 4 # 4 4 H = 0

Thus states of the box-ball system with $n$ colored balls become elements in an (semi) infinite tensor product of $B^{1,1}$ in type $A_n^{(1)}$ where only a finite number of factors are not equal to $1$. The time evolution is given by passing through a carrier, a factor $B^{1,s}$ with $s \gg 1$, using a series of combinatorial $R$-matrices with the initial element for the carrier being $1^s$.

v,a,b,c = list(crystals.KirillovReshetikhin(['A',3,1], 1,1))
carrier = crystals.KirillovReshetikhin(['A',3,1], 1, 5)
initial = [v,v,v,v,a,b,v,a,b,c]
latex.eval(evolve_state_latex(initial, carrier))

''

BBS = SolitonCellularAutomata('1111111313452451134411', ['A',4,1])
l = BBS.latex_states_word(2)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 {\color{gray} 1} 3 4 5 2 4 5 {\color{gray} 1} {\color{gray} 1} 3 4 4 {\color{gray} 1} {\color{gray} 1} \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 4 5 {\color{gray} 1} 5 {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} 3 4 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

v,a,b,c,d = list(crystals.KirillovReshetikhin(['A',4,1], 1,1))
carrier = crystals.KirillovReshetikhin(['A',4,1], 1, 5)
initial = [v,v,v,v,v,v,v,b,v,b,c,d,a,c,d,v,v,c,d,d,v,v]
next = evolve_state(initial, carrier, True)

1 1 1 1 1 1 | | | | | | 11111 --+-- 11111 --+-- 11111 --+-- 11111 --+-- 11113 --+-- 11133 --+-- 11334 | | | | | | 1 1 1 3 3 4 1 3 1 3 4 5 | | | | | | --+-- 13345 --+-- 11345 --+-- 13455 --+-- 11455 --+-- 11255 --+-- 11245 | | | | | | 5 1 5 1 2 4 2 4 5 1 1 4 | | | | | | --+-- 11145 --+-- 11115 --+-- 11114 --+-- 11145 --+-- 11455 --+-- 11155 | | | | | | 1 1 4 5 5 1 5 5 1 1 | | | | --+-- 11115 --+-- 11111 --+-- 11111 --+-- 11111 | | | | 1 1 1 1

''.join(repr(x).strip('[]') for x in initial)
''.join(repr(x).strip('[]') for x in next)

'1111111313452451145511' '1113345151241145511111'

Three more properties of box-ball systems we want to note are:

1. Solitons of length $s$ are parameterized by crystals $B^{1,s}$ of type $A_{n-1}^{(1)}$ given by subtracting 1 from each entry.
2. The scattering of two solitons corresponds to the combinatorial $R$-matrix of type $A_{n-1}^{(1)}$.
3. The phase shift, the difference of positions if there was no interaction, is equal to $2 \ell - H(x)$, where $\ell$ is the length of the smaller soliton.

SCA = SolitonCellularAutomata('111241111355', ['A',4,1])
l = SCA.latex_states_word(10)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 5 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} {\color{gray} 1} 3 5 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 {\color{gray} 1} 3 5 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 5 {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 5 {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 5 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

K = crystals.KirillovReshetikhin(['A',3,1], 1, 2)
Kp = crystals.KirillovReshetikhin(['A',3,1], 1, 3)
R = K.R_matrix(Kp)
H = K.local_energy_function(Kp)
T = R.domain()
x = T(K(rows=[[1,3]]), Kp(rows=[[2,4,4]]))
ascii_art(x, '   |--->   ', R(x), '       H = ', H(x))

1 3 # 2 4 4 |---> 1 3 4 # 2 4 H = 1

However, KR crystals $B^{r,s}$ (conjecturally) exists for all affine types, all nodes $r$ of the classical Dynkin diagram, and $s \in \ZZ_{>0}$. Moreover, there is a combinatorial $R$-matrix and vacuum states. We can use the same notation of states as being an (semi) infinite tensor product of $B^{r,1}$ and carriers $B^{r,s}$ for $s \gg 1$ with a series of combinatorial $R$-matrices being used to do time evolution. This appears to still exhibits solitonic behavior with the same properties as for the box-ball system. We call these dynamical systems soliton cellular automata (SCA).

ascii_art([ct.dynkin_diagram() for ct in CartanType.samples(affine=True)])

[ [ [ 0 [ O---------------+ O 0 [ | | | [ | | | [ O<=>O O---O---O---O---O O<=>O O---O---O---O=>=O O<=>O [ 0 1 1 2 3 4 5 0 1 1 2 3 4 5 0 1 [ A1~ , A5~ , B1~ , B5~ , C1~ , 0 O-------+ 0 O O 5 | | | | | | | | O=>=O---O---O---O=<=O O---O---O O---O---O---O 0 1 2 3 4 5 3 1 2 1 2 3 4 C5~ , D3~ , D5~ , O 0 | | O 2 O 2 | | | | O---O---O---O---O O---O---O---O---O---O---O 1 3 4 5 6 0 1 3 4 5 6 7 E6~ , E7~ , O 2 | | O---O---O=>=O---O 3 O---O---O---O---O---O---O---O 0 1 2 3 4 O=<=O---O 1 3 4 5 6 7 8 0 1 2 0 E8~ , F4~ , G2~ , O 0 | 4 | O=<=O O=<=O---O---O---O=<=O O---O---O---O=<=O 0 1 0 1 2 3 4 5 1 2 3 4 5 BC1~ , BC5~ , B5~* , O---O---O=<=O---O 3 4 O=<=O---O---O=>=O 0 1 2 3 4 O=>=O---O O=>=O 0 1 2 3 4 1 2 0 0 1 C4~* , F4~* , G2~* , BC1~* , ] ] ] ] ] ] O=>=O---O---O---O=>=O ] 0 1 2 3 4 5 ] BC5~* ]

SCA = SolitonCellularAutomata('111133111223', ['D',4,3])
l = SCA.latex_states_word(15)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 {\color{gray} 1} {\color{gray} 1} 2 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 {\color{gray} 1} 2 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 2 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 0 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 \overline{3} 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 0 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{2} \emptyset {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 \overline{1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 0 \emptyset {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 10 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 11 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 0 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 12 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 3 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 13 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 3 {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 14 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 3 {\color{gray} 1} {\color{gray} 1} 2 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

The first question is how do we determine a soliton. We do this by finding tensor products such that the sum of all of the local energy equals $H(f_r v \otimes v)$, where $v$ is the vacuum element. However, the notion of length is a little more subtle.

SCA = SolitonCellularAutomata([[1],[1],[2],[-3],[1],[1],[1],[3],[-1]], ['C',3,1])
show(SCA.latex_states_word(10))

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 \overline{1} \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} {\color{gray} 1} {\color{gray} 1} 3 \overline{1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} {\color{gray} 1} 3 \overline{1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} \overline{2} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} \overline{2} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} \overline{2} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

SCA = SolitonCellularAutomata([[-2],['E'],[1],[1],[1],[1],[2],[3],[-2],[1],[1],[1],[1],[2],[2],[2],[-3]], ['A',6,2])
show(SCA.latex_states_word(10))

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{2} \emptyset {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 \overline{3} \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{2} \emptyset {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{2} \emptyset {\color{gray} 1} {\color{gray} 1} 2 3 \overline{2} {\color{gray} 1} {\color{gray} 1} 2 2 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{2} \emptyset {\color{gray} 1} 2 3 \overline{2} {\color{gray} 1} 2 2 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{2} \emptyset 2 3 \overline{2} 2 2 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{1} \emptyset 3 \overline{2} 2 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{1} \emptyset {\color{gray} 1} {\color{gray} 1} 3 \overline{1} 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{1} \emptyset {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 \overline{1} {\color{gray} 1} 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{1} \emptyset {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 \overline{1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{1} \emptyset {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 \overline{1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

A rigged configuration is a sequence of partitions $\nu^{(r)}$, one for each node of the classical Dynkin diagram, with integers attached to row called riggings.

There is a (conjectural) bijection $\Phi$ between rigged configrations and tensor products of KR crystals. This bijection is recursively defined and very techincal (it comes from the Bethe ansatz of a related object in mathematical physics called Heisenberg spin chains), however, it linearizes the dynamics of SCA. Thus, it allows us to extract information about SCA and the time evolution.

---

Theorem: [Liu-S. 2017]
Assuming certain conjectures hold about the bijection $\Phi$, we show the following for certain classes of SCA:

1. Time evolution is described by changing the riggings on $\nu^{(r)}$ by adding their corresponding row length.
2. The lengths of the solitons corresponds to the rows of $\nu^{(r)}$.
3. Solitons of length $s$ are parameterized by crystals $B^{r',s}$ given by removing $r$ from the classical Dynkin diagram and $r'$ is the (set of) adjacent node(s).
4. The scattering of two solitons corresponds to the branched combinatorial $R$-matrix.
5. The phase shift is equal to $2 \ell + A_{rr'} H(x)$, where $\ell$ is the length of the smaller soliton and $A_{rr'}$ is the Cartan matrix.

SCA = SolitonCellularAutomata([[-4],[1],[1],[1],[4],[-3],[-2],[1],[1],[1],[1],[1],[3],[3],[-4],[-3],[1],[1],[1],[1],[1],[1],[2],[2],[2],[2],[3],[4]], ['D',4,1])
show(SCA.latex_states_word(10))

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{4} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 \overline{3} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 3 4 \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{4} {\color{gray} 1} 4 \overline{3} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{4} \overline{3} 4 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 3 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} 2 2 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{4} \overline{3} \overline{2} {\color{gray} 1} 4 {\color{gray} 1} 3 3 \overline{4} \overline{3} 2 2 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \overline{4} \overline{3} \overline{1} 2 3 3 4 \overline{4} \overline{3} 2 2 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 \overline{4} \overline{2} \overline{1} 4 \overline{3} {\color{gray} 1} {\color{gray} 1} 2 2 \overline{4} 3 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 \overline{4} \overline{2} \overline{2} {\color{gray} 1} {\color{gray} 1} 2 4 \overline{3} {\color{gray} 1} 2 2 3 \overline{4} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 \overline{4} \overline{2} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 \overline{3} {\color{gray} 1} 2 3 \overline{4} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 \overline{4} \overline{2} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 \overline{3} {\color{gray} 1} {\color{gray} 1} 2 3 \overline{4} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 3 \overline{4} \overline{2} \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 4 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 \overline{4} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 4 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

for st in SCA._states:
    print(' '.join(repr(x).strip('[]') for x in st))
    ascii_art(st.to_rigged_configuration())
    print("")

-4 1 1 1 4 -3 -2 1 1 1 1 1 3 3 -4 -3 1 1 1 1 1 1 2 2 2 2 3 4 11[ ][ ][ ][ ][ ][ ]-4 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 14[ ][ ][ ][ ]6 0[ ][ ][ ]0 0[ ]-1 0[ ]0 17[ ][ ][ ]14 -1[ ][ ]-1 24[ ]24 0[ ]0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -4 1 4 -3 -2 1 1 1 1 3 3 -4 -3 1 1 1 1 2 2 2 2 3 4 1 1 1 1 1 1 39[ ][ ][ ][ ][ ][ ]2 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 42[ ][ ][ ][ ]10 0[ ][ ][ ]0 0[ ]-1 0[ ]0 45[ ][ ][ ]17 -1[ ][ ]-1 52[ ]25 0[ ]0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -4 -3 4 -2 1 1 1 3 3 -4 -3 1 1 2 2 2 2 3 4 1 1 1 1 1 1 1 1 1 1 1 1 39[ ][ ][ ][ ][ ][ ]8 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 42[ ][ ][ ][ ]14 0[ ][ ][ ]0 0[ ]-1 0[ ]0 45[ ][ ][ ]20 -1[ ][ ]-1 52[ ]26 0[ ]0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -4 -3 -2 1 4 1 3 3 -4 -3 2 2 2 2 3 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 39[ ][ ][ ][ ][ ][ ]14 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 42[ ][ ][ ][ ]18 0[ ][ ][ ]0 0[ ]-1 0[ ]0 45[ ][ ][ ]23 -1[ ][ ]-1 52[ ]27 0[ ]0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -4 -3 -1 2 3 3 4 -4 -3 2 2 3 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 39[ ][ ][ ][ ][ ][ ]20 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 42[ ][ ][ ][ ]22 0[ ][ ][ ]0 0[ ]-1 0[ ]0 45[ ][ ][ ]26 -1[ ][ ]-1 52[ ]28 0[ ]0 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 -4 -2 -1 4 -3 1 1 2 2 -4 3 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 39[ ][ ][ ][ ][ ][ ]26 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 42[ ][ ][ ][ ]26 0[ ][ ][ ]0 0[ ]-1 0[ ]0 45[ ][ ][ ]29 -1[ ][ ]-1 52[ ]29 0[ ]0 1 1 1 1 1 1 1 2 2 3 -4 -2 -2 1 1 2 4 -3 1 2 2 3 -4 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 39[ ][ ][ ][ ][ ][ ]32 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 42[ ][ ][ ][ ]30 0[ ][ ][ ]0 0[ ]-1 0[ ]0 45[ ][ ][ ]32 -1[ ][ ]-1 52[ ]30 0[ ]0 1 2 2 3 -4 -2 -2 1 1 1 1 2 2 4 -3 1 2 3 -4 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 39[ ][ ][ ][ ][ ][ ]38 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 42[ ][ ][ ][ ]34 0[ ][ ][ ]0 0[ ]-1 0[ ]0 45[ ][ ][ ]35 -1[ ][ ]-1 52[ ]31 0[ ]0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 -4 -2 -2 1 1 1 1 1 1 2 2 4 -3 1 1 2 3 -4 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 67[ ][ ][ ][ ][ ][ ]44 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 70[ ][ ][ ][ ]38 0[ ][ ][ ]0 0[ ]-1 0[ ]0 73[ ][ ][ ]38 -1[ ][ ]-1 80[ ]32 0[ ]0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 -4 -2 -2 1 1 1 1 1 1 1 1 2 2 4 -3 1 1 1 2 3 -4 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 67[ ][ ][ ][ ][ ][ ]50 1[ ][ ][ ][ ][ ]0 0[ ][ ][ ][ ]-1 0[ ][ ][ ][ ]0 70[ ][ ][ ][ ]42 0[ ][ ][ ]0 0[ ]-1 0[ ]0 73[ ][ ][ ]41 -1[ ][ ]-1 80[ ]33 0[ ]0

1. Question: Is this the correct definition of soltions and length?

   We think so.

2. Open Problem: The description of the phase shift is more subtle when the length is not the number of factors. Moreover, our statement about the phase shift likely does not generalize. Make this statement precise and true for the general case.

Some more examples because they are fun to generate.

BBS = SolitonCellularAutomata([[2],[1],[1],[1],[1],[3],[0],[-2],[1],[1],[3],[-3],[1],[1],[1],[1],[1],[1],[2],[2],[2],[0],[-2]], ['B',4,1])
l = BBS.latex_states_word(13)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 0 \overline{2} {\color{gray} 1} {\color{gray} 1} 3 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 0 \overline{2} \\ t = 1 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 3 0 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 2 2 0 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 2 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} 0 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 \overline{3} 2 2 2 0 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 3 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 {\color{gray} 1} {\color{gray} 1} \overline{2} {\color{gray} 1} 2 2 3 \overline{3} \overline{3} 3 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 4 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 {\color{gray} 1} {\color{gray} 1} 2 3 \overline{1} \overline{3} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 5 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 \overline{4} {\color{gray} 1} 2 4 \overline{3} \overline{2} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 6 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} 2 4 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 3 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 7 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} {\color{gray} 1} 3 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 8 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 {\color{gray} 1} 3 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 9 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 10 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 11 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ t = 12 & \cdots {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 0 \overline{4} \overline{3} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 4 \overline{2} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 2 3 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} 0 {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} {\color{gray} 1} \\ \end{array}$$

BBS = SolitonCellularAutomata([[-1,2],[2,1],[2,1],[2,1],[2,1],[-2,2],[-1,2],[2,1],[2,1],[-2,1],[-2,1],[-2,2]], ['C',2,1], vacuum=2)
l = BBS.latex_states_word(11)
show(l)

$$\begin{array}{c|c} t = 0 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{2}\end{aligned} \\ t = 1 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 2 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 3 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}\overline{2}\\\overline{1}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 4 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}\overline{2}\\\overline{1}\end{aligned} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}\overline{2}\\\overline{1}\end{aligned} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 5 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}\overline{2}\\\overline{1}\end{aligned} \begin{aligned}\overline{2}\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 6 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} \begin{aligned}\overline{2}\\\overline{1}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 7 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 8 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 9 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ t = 10 & \cdots {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} \begin{aligned}2\\\overline{1}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}1\\\overline{2}\end{aligned} \begin{aligned}1\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \begin{aligned}2\\\overline{2}\end{aligned} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} {\color{gray} \begin{aligned}1\\2\end{aligned}} \\ \end{array}$$