poincarePolynomial -- computes the Poincaré polynomial of a ranked poset with a unique minimal element

Synopsis

• Usage:

  p = poincarePolynomial P

  p = poincarePolynomial(P, VariableName => symbol)

  p = poincare P

• Inputs:
  • P, an instance of the type Poset, a ranked poset
• Optional inputs:
  • VariableName => a symbol, default value t
• Outputs:
  • p, a ring element, the Poincaré polynomial of $P$

Description

i1 : n = 5;

i2 : factor poincarePolynomial booleanLattice n

            5
o2 = (t + 1)

o2 : Expression of class Product

i3 : R = QQ[x,y,z];

i4 : A = {x,y,z,x+y,x+z,y+z,x-y,x-z,y-z};

i5 : LA = intersectionLattice(A, R);

i6 : factor poincarePolynomial LA

o6 = (t + 1)(3t + 1)(5t + 1)

o6 : Expression of class Product

See also

• intersectionLattice -- generates the intersection lattice of a hyperplane arrangement
• isRanked -- determines if a poset is ranked
• moebiusFunction -- computes the Moebius function at every pair of elements of a poset
• rankFunction -- computes the rank function of a ranked poset