1414.sagews

P=Poset([[1,2],[3],[3,4],[6],[5],[6],[]])
P.plot(vertex_colors='white')

P.linear_extensions()

[[0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 4, 3, 5, 6], [0, 1, 2, 4, 5, 3, 6], [0, 2, 1, 3, 4, 5, 6], [0, 2, 1, 4, 3, 5, 6], [0, 2, 1, 4, 5, 3, 6], [0, 2, 4, 1, 3, 5, 6], [0, 2, 4, 1, 5, 3, 6], [0, 2, 4, 5, 1, 3, 6]]

LP=P.linear_extensions()
n=P.size()
QX=PolynomialRing(QQ,'x',n)
A=0
for i in LP:
    A=A+1/prod( QX.gen(int(i[j-1].n())) - QX.gen(int(i[j].n())) for j in range(1,n))
A.factor()

(-1) * (-x5 + x6)^-1 * (-x4 + x5)^-1 * (-x3 + x6)^-1 * (-x2 + x4)^-1 * (x2 - x3)^-1 * (x1 - x3)^-1 * (-x0 + x2)^-1 * (-x0 + x1)^-1 * (x2 - x6) * (-x0 + x3)

P2=Poset([[1,2],[3],[3],[]])
LP2=P2.linear_extensions()
n=P2.size()
QX=PolynomialRing(QQ,'x',n)
A=0
for i in LP2:
    A=A+1/prod( QX.gen(int(i[j-1].n())) - QX.gen(int(i[j].n())) for j in range(1,n))
print(A.factor())
P2.plot(vertex_colors='white')

(x2 - x3)^-1 * (x1 - x3)^-1 * (-x0 + x2)^-1 * (-x0 + x1)^-1 * (x0 - x3)

P3=Poset([[1,2],[3,4],[3,4],[],[]])
P3.plot(vertex_colors='white')

LP3=P3.linear_extensions()
n=P3.size()
QX=PolynomialRing(QQ,'x',n)
A=0
for i in LP3:
    A=A+1/prod( QX.gen(int(i[j-1].n())) - QX.gen(int(i[j].n())) for j in range(1,n))
A.factor()

P4=Poset([[1,2],[3,4],[4,5],[6],[6,7],[7]])
P4.plot(vertex_colors='white')