Lab3-Mobius_order_polynomials.sagews

Mobius Functions, Order Polynomials, and Linear Extensions

Math 737 - Lab 3

Sage can compute the Mobius function of a poset. Create a poset and then use the commands moebius_function and moebius_function_matrix to determine how they work and how they relate to Mobius function computations done by hand.

P = Posets.BooleanLattice(3)
P.plot()
P.moebius_function(0,4)
P.moebius_function_matrix()

Exercise 1: Use the above commands to compute the Mobius function of some of our example posets P, Q, R, etc. Now compute the Mobius function of the product of these posets. What fact about Mobius functions does this show you?

Exercise 2: Compute the Mobius function of the Boolean lattice for n=3,4. What do you notice? Do the same for the strong Bruhat order.

Exercise 3: Compute the order polynomial and the number of linear extensions of the following posets:

- An antichain of 5 elements

- A chain of 5 elements

- A poset that is the product of two chains of 3 and 5 elements

To find the number of linear extensions, you may find it helpful to first find the command to generate all linear extensions and then put the whole statement inside the command len() which finds the length of this list.

#Example of using len
b = [2,4,6,8,10]
len(b)

5

P = Posets.

Exercise 4: Compute the order polynomial of the product of two chains of a and b elements for more (small) values of a and b. Factor these polynomials. Can you conjecture a formula for these polynomials?

Similarly, compute the number of linear extensions of the product of two chains of a and b elements for more (small) values of a and b. Factor these numbers. Can you conjecture a counting formula?

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