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def rectangle(a,b):
    return Poset(([(i+1,j+1) for i in range(a) for j in range(b)],lambda x, y: x[0] <= y[0] and x[1] <= y[1]))

def shifted_staircase(n):
    return Poset(([(i+1,j+1) for i in range(n) for j in range(n) if i <= j ],lambda x, y: x[0] <= y[0] and x[1] <= y[1]))

def double_tailed_diamond(n):
    x={i:[i+1] for i in range(1,n-1)}
    y={n-1:[n,n+1],n:[n+2]}
    z={i:[i+1] for i in range(n+1,2*n)}
    z.update(y)
    z.update(x)
    return Poset(z)

def e6_minuscule_poset():
    return Poset(([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], [[1,2],[2,3],[3,4],[4,5],[3,6],[4,7],[5,8],[6,7],[7,8],[7,9],[8,10],[9,10],[10,11],[9,12],[10,13],[11,14],[12,13],[13,14],[14,15],[15,16]]), cover_relations = True)

def e7_minuscule_poset():
    return Poset(([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27], [[1,2],[2,3],[3,4],[4,5],[5,6],[4,7],[5,8],[6,9],[7,8],[8,9],[8,10],[9,11],[10,11],[11,12],[10,13],[11,14],[12,15],[13,14],[14,15],[15,16],[16,17],[13,18],[14,19],[15,20],[16,21],[17,22],[18,19],[19,20],[20,21],[21,22],[21,23],[22,24],[23,24],[24,25],[25,26],[26,27]]), cover_relations = True)

def type_a_root_poset(n):
    return Poset(([(i+1,j+1) for i in range(n) for j in range(n) if i + j >= n-1],lambda x, y: x[0] <= y[0] and x[1] <= y[1]))

def type_b_root_poset(n):
    return Poset(([(i+1,j+1) for i in range(2*n-1) for j in range(2*n-1) if i + j >= 2*n-2 and i <= j],lambda x, y: x[0] <= y[0] and x[1] <= y[1]))

def h3_root_poset():
    return Poset({1:[4],2:[4,5],3:[5],4:[6,7],5:[7],6:[8,9],7:[9],8:[10],9:[10,11],10:[12],11:[12],12:[13],13:[14],14:[15]})

def i2_root_poset(m):
    x={i:[i+1] for i in range(m)}
    y={m+1:[1]}
    y.update(x)
    return Poset(y)

def trapezoid(a,b):
    return Poset(([(i+1,j+1) for i in range(a) for j in range(a+b) if i + j >= a-1 and j <= b+i-1],lambda x, y: x[0] <= y[0] and x[1] <= y[1]))

def chain_of_vs(n):
    return Poset({1:[2,3]}).product(posets.ChainPoset(n))

def diagram_poset(par):
    return Partition(par).cell_poset()

def shifted_diagram_poset(par):
    els = []
    for i in range(len(par)):
        for j in range(par[i]):
            els += [(i,i+j)]
    return Poset([els, lambda x,y: x[0] <= y[0] and x[1] <= y[1]])

def q_signed_tog_vecs(P,q):
    L = list(P.order_ideals_lattice())
    vecs = []
    for p in P:
        vec = [0 for i in range(len(L))]
        for i in range(len(L)):
            I = L[i]
            if P.order_ideal_toggle(I,p) != I and P.order_ideal_toggle(I,p).issuperset(I):
                vec[i] = 1
            if P.order_ideal_toggle(I,p) != I and P.order_ideal_toggle(I,p).issubset(I):
                vec[i] = -q
        vecs += [vec]
    return vecs

def signed_tog_vecs(P):
    return q_signed_tog_vecs(P,1)

def constant_vec(P):
    return [1 for I in P.order_ideals_lattice()]

def antichain_cardinality_vec(P):
    L = list(P.order_ideals_lattice())
    vec = [0 for i in range(len(L))]
    for p in P:
        for i in range(len(L)):
            I = L[i]
            if P.order_ideal_toggle(I,p) != I and P.order_ideal_toggle(I,p).issubset(I):
                vec[i] += 1
    return vec

def order_ideal_cardinality_vec(P):
    L = list(P.order_ideals_lattice())
    vec = [L[i].cardinality() for i in range(len(L))]
    return vec

def rank_alternating_order_ideal_cardinality_vec(P):
    L = list(P.order_ideals_lattice())
    vec = [0 for i in range(len(L))]
    for p in P:
        for i in range(len(L)):
            I = L[i]
            if p in I:
                vec[i] += (-1)^(P.rank(p))
    return vec

def check_vec_in_tog_space(P,vec,verbose=false):
    vecs = signed_tog_vecs(P)
    vecs += [constant_vec(P)]
    M = Matrix(vecs, base_ring=QQ).transpose()
    if verbose:
        M.plot().show()
        print("")
    v = vector(vec)
    if verbose:
        print(v)
        print("")
    if verbose:
        if v in M.column_space():
            print(M.solve_right(v))
    return v in M.column_space()

def check_ac_in_tog_space(P,verbose=false):
    return check_vec_in_tog_space(P,antichain_cardinality_vec(P),verbose)

def check_oic_in_tog_space(P,verbose=false):
    return check_vec_in_tog_space(P,order_ideal_cardinality_vec(P),verbose)

def check_raoic_in_tog_space(P,verbose=false):
    return check_vec_in_tog_space(P,rank_alternating_order_ideal_cardinality_vec(P),verbose)

def check_vec_in_q_tog_space(P,vec,verbose=false):
    R.<q> = PolynomialRing(QQ)
    vecs = q_signed_tog_vecs(P,q)
    vecs += [constant_vec(P)]
    M = Matrix(vecs, base_ring=Frac(R)).transpose()
    v = vector(vec)
    if verbose:
        print(v)
        print("")
    if verbose:
        if v in M.column_space():
            print(M.solve_right(v))
    return v in M.column_space()

def check_ac_in_q_tog_space(P,verbose=false):
    return check_vec_in_q_tog_space(P,antichain_cardinality_vec(P),verbose)

def antichain_vecs(P):
    L = list(P.order_ideals_lattice())
    vecs = []
    for p in P:
        vec = [0 for i in range(len(L))]
        for i in range(len(L)):
            I = L[i]
            if P.order_ideal_toggle(I,p) != I and P.order_ideal_toggle(I,p).issubset(I):
                vec[i] = 1
        vecs += [vec]
    return vecs

def order_ideal_vecs(P):
    L = list(P.order_ideals_lattice())
    vecs = []
    for p in P:
        vec = [0 for i in range(len(L))]
        for i in range(len(L)):
            I = L[i]
            if p in I:
                vec[i] = 1
        vecs += [vec]
    return vecs

def antichain_space(P):
    vecs = signed_tog_vecs(P)
    vecs += [constant_vec(P)]
    M = Matrix(vecs, base_ring=QQ).transpose()
    vecs = antichain_vecs(P)
    N = Matrix(vecs, base_ring=QQ).transpose()
    return M.column_space().intersection(N.column_space())

def order_ideal_space(P):
    vecs = signed_tog_vecs(P)
    vecs += [constant_vec(P)]
    M = Matrix(vecs, base_ring=Frac(QQ)).transpose()
    vecs = order_ideal_vecs(P)
    N = Matrix(vecs, base_ring=Frac(QQ)).transpose()
    return M.column_space().intersection(N.column_space())

def q_antichain_space(P,q):
    vecs = q_signed_tog_vecs(P,q)
    vecs += [constant_vec(P)]
    M = Matrix(vecs, base_ring=Frac(parent(q))).transpose()
    vecs = antichain_vecs(P)
    N = Matrix(vecs, base_ring=Frac(parent(q))).transpose()
    return M.column_space().intersection(N.column_space())

def q_order_ideal_space(P,q):
    vecs = q_signed_tog_vecs(P,q)
    vecs += [constant_vec(P)]
    M = Matrix(vecs, base_ring=Frac(parent(q))).transpose()
    vecs = order_ideal_vecs(P)
    N = Matrix(vecs, base_ring=Frac(parent(q))).transpose()
    return M.column_space().intersection(N.column_space())

def approx_q_antichain_space(P):
    V = antichain_space(P)
    for i in range(10):
        V = V.intersection(q_antichain_space(P,Integer(randint(1,100)))) #this approximates generic q by taking the intersection over q being several random integers
    return V

def approx_q_order_ideal_space(P):
    V = order_ideal_space(P)
    for i in range(10):
        V = V.intersection(q_order_ideal_space(P,Integer(randint(1,100)))) #this approximates generic q by taking the intersection over q being several random integers
    return V

# "good" examples

P = rectangle(2,2)
P.show()
check_ac_in_tog_space(P,verbose=true)
(0, 1, 1, 1, 2, 1) (-1, -1/2, -1/2, 0, 1) True 
P = rectangle(2,2)
P.show()
check_ac_in_q_tog_space(P,verbose=true)
(0, 1, 1, 1, 2, 1) ((-q - 1)/(q^2 + 1), -q/(q^2 + 1), -q/(q^2 + 1), (-q + 1)/(q^2 + 1), (q + 1)/(q^2 + 1)) True 
P = double_tailed_diamond(4)
P.show()
check_ac_in_q_tog_space(P,verbose=true)
(0, 1, 1, 1, 1, 1, 2, 1, 1, 1) ((-q^3 - 1)/(q^4 + 1), (-q^3 - q)/(q^4 + 1), (-q^3 - q^2)/(q^4 + 1), -q^3/(q^4 + 1), -q^3/(q^4 + 1), (-q^3 + 1)/(q^4 + 1), (-q^3 + q)/(q^4 + 1), (-q^3 + q^2)/(q^4 + 1), (q^3 + 1)/(q^4 + 1)) True 
P = rectangle(3,4)
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True True True True Vector space of degree 35 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 -1 -1 -1 -1 -1 0 -1 -1 -1 -1 -2 -1 -1 -2 -1 -1 -1 -2 -2 -1] [ 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 0 0] [ 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0] [ 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 1 0] [ 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1] [ 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1] Vector space of degree 35 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 -1 0 -1 0 0 -1 0 -1 -1 0 0 -1 0 -1 -1 -1 0 -1 -1 -1 -2 -1 -1 -2 -1 -1 -1 -2 -1 -2] [ 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 0 0 1] [ 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1] [ 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1] [ 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1] [ 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1] Vector space of degree 35 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 -1 -1 -1 -1 -1 0 -1 -1 -1 -1 -2 -1 -1 -2 -1 -1 -1 -2 -2 -1] [ 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 0 0] [ 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0] [ 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 1 0] [ 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1] [ 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1] Vector space of degree 35 and dimension 2 over Rational Field Basis matrix: [0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1] 
P = shifted_staircase(4)
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True True True True Vector space of degree 16 and dimension 7 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 0 -1 -1 0 -1 0 -2] [ 0 0 1 0 0 0 0 0 0 0 -1 -1 0 -1 0 -2] [ 0 0 0 1 0 0 1 0 0 0 2 2 -1 1 0 2] [ 0 0 0 0 1 0 1 1 0 0 2 2 0 2 0 3] [ 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 2] [ 0 0 0 0 0 0 0 0 1 1 -1 -1 1 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 16 and dimension 7 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 0 -1 -1 0 -1 0 -2] [ 0 0 1 0 0 0 -1 -1 0 0 0 0 -1 -1 0 -1] [ 0 0 0 1 0 0 1 0 0 -1 2 1 0 2 0 2] [ 0 0 0 0 1 0 1 1 0 0 2 2 0 2 0 3] [ 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 2] [ 0 0 0 0 0 0 0 0 1 1 -1 -1 1 -1 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 16 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 -1 -1 0 0 -1 -1 -1 0] [ 0 0 1 0 0 0 0 0 0 0 -1 -1 0 -1 -1 -1] [ 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0] [ 0 0 0 0 1 0 1 1 1 1 1 1 1 2 1 1] [ 0 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1] Vector space of degree 16 and dimension 2 over Rational Field Basis matrix: [0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 1] [0 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1] 
P = double_tailed_diamond(5)
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True True True True Vector space of degree 12 and dimension 9 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 0 0 -1] [ 0 0 1 0 0 0 0 0 0 0 0 -1] [ 0 0 0 1 0 0 0 0 0 0 0 -1] [ 0 0 0 0 1 0 0 0 0 0 0 -1] [ 0 0 0 0 0 1 0 1 0 0 0 4] [ 0 0 0 0 0 0 1 1 0 0 0 4] [ 0 0 0 0 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 12 and dimension 9 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 0 0 -1] [ 0 0 1 0 0 0 0 0 0 0 0 -1] [ 0 0 0 1 0 0 0 0 0 0 0 -1] [ 0 0 0 0 1 0 0 -1 0 0 0 0] [ 0 0 0 0 0 1 0 1 0 0 0 4] [ 0 0 0 0 0 0 1 1 0 0 0 4] [ 0 0 0 0 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 12 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 -1 0 0 0] [ 0 0 1 0 0 0 0 0 0 -1 0 0] [ 0 0 0 1 0 0 0 0 0 0 -1 0] [ 0 0 0 0 1 0 0 0 0 0 0 -1] [ 0 0 0 0 0 1 0 1 1 1 1 1] [ 0 0 0 0 0 0 1 1 1 1 1 1] Vector space of degree 12 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 -1 0 0 0] [ 0 0 1 0 0 0 0 0 0 -1 0 0] [ 0 0 0 1 0 0 0 0 0 0 -1 0] [ 0 0 0 0 0 1 0 1 1 1 1 1] [ 0 0 0 0 0 0 1 1 1 1 1 1] 
P = e6_minuscule_poset()
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True True True True Vector space of degree 27 and dimension 11 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 0 0 1 0 1 0 0 0 0 0 3 0 3 -1 -1 2 0 3 -2 1 3 1 0 0 4] [ 0 0 0 0 0 1 1 0 1 0 0 0 3 0 3 0 0 3 0 3 -1 2 3 2 0 0 6] [ 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 2 2 0 2 0 0 4] [ 0 0 0 0 0 0 0 0 0 1 1 0 -1 0 -1 1 1 0 0 -1 1 0 -1 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 -1 0 1 -1 0 -1 2 1 -2 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 27 and dimension 11 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 0 0 -3] [ 0 0 0 1 0 0 -1 0 -1 0 0 0 0 0 0 -1 -1 -1 0 0 -1 -1 0 -1 0 0 -2] [ 0 0 0 0 1 0 1 0 0 0 -1 0 3 -1 2 0 0 3 0 3 -2 1 3 1 0 0 4] [ 0 0 0 0 0 1 1 0 1 0 0 0 3 0 3 0 0 3 0 3 -1 2 3 2 0 0 6] [ 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 2 2 0 2 0 0 4] [ 0 0 0 0 0 0 0 0 0 1 1 0 -1 0 -1 1 0 -1 0 -1 2 1 -2 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 -1 0 1 -1 0 -2 2 0 -1 1 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 -1 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 27 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 0 0 -1 -1 -1 0 0] [ 0 0 1 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 -1 -1 -1 0 -1 -1 -1 0] [ 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 -1 0 0 -1 0 -1 0 -1 -1 -1 -1 -1 -1] [ 0 0 0 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 2 0 1 1 1 1 1 0] [ 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 2 2 1 2 1 2 1 2 2 1 1] [ 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 2 1 1 1] Vector space of degree 27 and dimension 1 over Rational Field Basis matrix: [0 1 1 0 0 1 1 1 2 0 1 0 1 1 2 1 1 2 0 1 1 2 1 2 1 1 2] 
P = e7_minuscule_poset()
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True True True True Vector space of degree 56 and dimension 17 over Rational Field Basis matrix: 17 x 56 dense matrix over Rational Field Vector space of degree 56 and dimension 17 over Rational Field Basis matrix: 17 x 56 dense matrix over Rational Field Vector space of degree 56 and dimension 7 over Rational Field Basis matrix: 7 x 56 dense matrix over Rational Field Vector space of degree 56 and dimension 1 over Rational Field Basis matrix: 1 x 56 dense matrix over Rational Field 
P = type_a_root_poset(5)
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True False True False Vector space of degree 132 and dimension 5 over Rational Field Basis matrix: 5 x 132 dense matrix over Rational Field Vector space of degree 132 and dimension 5 over Rational Field Basis matrix: 5 x 132 dense matrix over Rational Field Vector space of degree 132 and dimension 1 over Rational Field Basis matrix: 1 x 132 dense matrix over Rational Field Vector space of degree 132 and dimension 0 over Rational Field Basis matrix: 0 x 132 dense matrix over Rational Field 
P = type_b_root_poset(3)
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True False False False Vector space of degree 20 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1] [ 0 0 1 0 1 0 1 0 1 1 0 1 1/2 1/2 1/2 3/2 -1/2 1/2 0 1] [ 0 0 0 1 0 1 1 0 1 0 1 1 1/2 1/2 1/2 1/2 1/2 1/2 0 1] [ 0 0 0 0 0 0 0 1 0 -1 1 -1 0 0 1 -1 1 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 20 and dimension 5 over Rational Field Basis matrix: [ 0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1] [ 0 0 1 0 1 0 1 0 1 1 0 1 1/2 1/2 -1/2 1/2 1/2 3/2 0 1] [ 0 0 0 1 0 1 1 0 1 0 1 1 1/2 1/2 1/2 1/2 1/2 1/2 0 1] [ 0 0 0 0 0 0 0 1 0 -1 1 -1 0 0 1 -1 1 -1 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 20 and dimension 2 over Rational Field Basis matrix: [ 0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1] [ 0 0 1 -1 1 -1 0 0 0 1 -1 0 0 0 0 1 -1 0 0 0] Vector space of degree 20 and dimension 1 over Rational Field Basis matrix: [0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1] 
P = i2_root_poset(5)
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True False False False Vector space of degree 9 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 1 0 0 0 0 5/2] [ 0 0 1 1 0 0 0 0 5/2] [ 0 0 0 0 1 0 0 0 -1] [ 0 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 1 -1] Vector space of degree 9 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 1 0 0 0 0 5/2] [ 0 0 1 1 0 0 0 0 5/2] [ 0 0 0 0 1 0 0 0 -1] [ 0 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 1 -1] Vector space of degree 9 and dimension 1 over Rational Field Basis matrix: [ 0 1 -1 0 0 0 0 0 0] Vector space of degree 9 and dimension 1 over Rational Field Basis matrix: [ 0 1 -1 0 0 0 0 0 0] 
P = h3_root_poset()
P.show()
check_ac_in_tog_space(P)
check_oic_in_tog_space(P)
check_raoic_in_tog_space(P)
check_ac_in_q_tog_space(P)
print("")
antichain_space(P)
order_ideal_space(P)
approx_q_antichain_space(P)
approx_q_order_ideal_space(P)
True False False False Vector space of degree 32 and dimension 9 over Rational Field Basis matrix: [ 0 1 0 0 0 1 1 0 1 1 0 0 0 3/2 0 3/2 0 0 3/2 0 0 3/2 0 3/2 -1 1/2 3/2 1/2 0 0 0 2] [ 0 0 1 0 1 0 1 1/2 1 1/2 0 0 0 3/2 0 3/2 1/2 1/2 2 -1/2 -1/2 1 0 3/2 -1 1/2 3/2 1/2 0 0 0 2] [ 0 0 0 1 1 1 0 1/2 1 1/2 0 0 1 0 1 1 1/2 1/2 1/2 1/2 1/2 1/2 0 0 1 1 0 1 0 0 0 2] [ 0 0 0 0 0 0 0 0 0 0 1 0 1 -1 0 -1 1 0 -1 1 1 0 0 -1 1 0 -1 0 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 1 -1 0 1 -1 0 1 -1 0 -1 2 1 -2 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 0 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 32 and dimension 9 over Rational Field Basis matrix: [ 0 1 0 0 0 1 1 0 1 1 0 0 0 3/2 0 3/2 0 0 3/2 0 0 3/2 0 3/2 -1 1/2 3/2 1/2 0 0 0 2] [ 0 0 1 0 1 0 1 1/2 1 1/2 0 0 0 3/2 0 3/2 -1/2 -1/2 1 1/2 1/2 2 0 3/2 -1 1/2 3/2 1/2 0 0 0 2] [ 0 0 0 1 1 1 0 1/2 1 1/2 0 0 1 0 1 1 1/2 1/2 1/2 1/2 1/2 1/2 0 0 1 1 0 1 0 0 0 2] [ 0 0 0 0 0 0 0 0 0 0 1 0 1 -1 0 -1 1 0 -1 1 0 -1 0 -1 2 1 -2 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 1 -1 0 1 -1 0 1 -1 0 -2 2 0 -1 1 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 -1 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] Vector space of degree 32 and dimension 1 over Rational Field Basis matrix: [ 0 1 -1 1 0 2 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1] Vector space of degree 32 and dimension 0 over Rational Field Basis matrix: [] 
## "bad" examples

P =  RootSystem(['D',4]).root_poset(facade=true)
P.show()
check_ac_in_tog_space(P)
L = list(P.order_ideals_lattice())
[sum(antichain_cardinality_vec(P)[L.index(I)] for I in O)/len(O) for O in P.rowmotion_orbits()]
False [2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 
P = trapezoid(3,4)
P.show()
check_ac_in_tog_space(P)
L = list(P.order_ideals_lattice())
[sum(antichain_cardinality_vec(P)[L.index(I)] for I in O)/len(O) for O in P.rowmotion_orbits()]
False [12/7, 12/7, 12/7, 12/7, 12/7] 
P = trapezoid(4,2)
P.show()
check_ac_in_tog_space(P,verbose=true)
L = list(P.order_ideals_lattice())
[sum(antichain_cardinality_vec(P)[L.index(I)] for I in O)/len(O) for O in P.rowmotion_orbits()]
(0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1) (-1/2, -1/2, 0, -1/2, 0, 0, 1/2, 1, 1/2, 3/2) True [3/2, 3/2, 3/2, 3/2] 
P = trapezoid(2,4)
P.show()
check_ac_in_tog_space(P,verbose=true)
L = list(P.order_ideals_lattice())
[sum(antichain_cardinality_vec(P)[L.index(I)] for I in O)/len(O) for O in P.rowmotion_orbits()]
/ext/sage/10.6/local/var/lib/sage/venv-python3.12.5/lib/python3.12/site-packages/scikits/__init__.py:1: DeprecationWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html __import__("pkg_resources").declare_namespace(__name__) /ext/sage/10.6/local/var/lib/sage/venv-python3.12.5/lib/python3.12/site-packages/scikits/__init__.py:1: DeprecationWarning: Deprecated call to `pkg_resources.declare_namespace('scikits')`. Implementing implicit namespace packages (as specified in PEP 420) is preferred to `pkg_resources.declare_namespace`. See https://setuptools.pypa.io/en/latest/references/keywords.html#keyword-namespace-packages __import__("pkg_resources").declare_namespace(__name__)
(0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1) False [4/3, 4/3, 4/3] 
P = trapezoid(1,3)
P.show()
check_ac_in_tog_space(P)
L = list(P.order_ideals_lattice())
[sum(antichain_cardinality_vec(P)[L.index(I)] for I in O)/len(O) for O in P.rowmotion_orbits()]
True [3/4] 
P = trapezoid(3,3)
P.show()
check_ac_in_tog_space(P)
L = list(P.order_ideals_lattice())
[sum(antichain_cardinality_vec(P)[L.index(I)] for I in O)/len(O) for O in P.rowmotion_orbits()]
True [3/2, 3/2, 3/2, 3/2] 
P = chain_of_vs(5)
P.show()
check_ac_in_tog_space(P)
L = list(P.order_ideals_lattice())
[sum(antichain_cardinality_vec(P)[L.index(I)] for I in O)/len(O) for O in P.rowmotion_orbits()]
False [15/7, 15/7, 15/7, 15/7, 15/7, 15/7, 15/7, 15/7] 
#checking for other posets with antichain cardinality in q-toggle space

for n in range(21):
    print("n="+str(n))
    for p in Partitions(n):
        if check_ac_in_q_tog_space(diagram_poset(p)):
            print(p)
    print()
n=0 [] n=1 [1] n=2 [2] [1, 1] n=3 [3] [1, 1, 1] n=4 [4] [2, 2] [1, 1, 1, 1] n=5 [5] [1, 1, 1, 1, 1] n=6 [6] [3, 3] [2, 2, 2] [1, 1, 1, 1, 1, 1] n=7 [7] [1, 1, 1, 1, 1, 1, 1] n=8 [8] [4, 4] [2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 1, 1] n=9 [9] [3, 3, 3] [1, 1, 1, 1, 1, 1, 1, 1, 1] n=10 [10] [5, 5] [2, 2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=11 [11] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=12 [12] [6, 6] [4, 4, 4] [3, 3, 3, 3] [2, 2, 2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=13 [13] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=14 [14] [7, 7] [2, 2, 2, 2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=15 [15] [5, 5, 5] [3, 3, 3, 3, 3] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=16 [16] [8, 8] [4, 4, 4, 4] [2, 2, 2, 2, 2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=17 [17] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=18 [18] [9, 9] [6, 6, 6] [3, 3, 3, 3, 3, 3] [2, 2, 2, 2, 2, 2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=19 [19] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] n=20 [20] [10, 10] [5, 5, 5, 5] [4, 4, 4, 4, 4] [2, 2, 2, 2, 2, 2, 2, 2, 2, 2] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 
for n in range(22):
    print("n="+str(n))
    for p in Partitions(n,max_slope=-1):
        if check_ac_in_q_tog_space(shifted_diagram_poset(p)):
            print(p)
    print()
n=0 [] n=1 [1] n=2 [2] n=3 [3] [2, 1] n=4 [4] n=5 [5] n=6 [6] [3, 2, 1] n=7 [7] n=8 [8] n=9 [9] n=10 [10] [4, 3, 2, 1] n=11 [11] n=12 [12] n=13 [13] n=14 [14] n=15 [15] [5, 4, 3, 2, 1] n=16 [16] n=17 [17] n=18 [18] n=19 [19] n=20 [20] n=21 [21] [6, 5, 4, 3, 2, 1]