zetaPolynomial -- computes the zeta polynomial of a poset

Synopsis

Usage:

z = zetaPolynomial P

z = zetaPolynomial(P, VariableName => symbol)
Inputs:
- P, an instance of the type Poset,
Optional inputs:
- VariableName => a symbol, default value q
Outputs:
- z, a ring element, the zeta polynomial of $P$

Description

The zeta polynomial of $P$ is the polynomial $z$ such that for every $i > 1$, $z(i)$ is the number of weakly increasing chains of $i-1$ vertices in $P$.

The zeta polynomial of the $n$ booleanLattice is $q^n$.

i1 : B = booleanLattice 3;

i2 : z = zetaPolynomial B

      3
o2 = q

o2 : QQ[q]

Thus, $z(2)$ is the number of vertices of $P$, and $z(3)$ is the number of total relations in $P$.

i3 : #B.GroundSet == sub(z, (ring z)_0 => 2)

o3 = true

i4 : #allRelations B == sub(z, (ring z)_0 => 3)

o4 = true

Ways to use zetaPolynomial :

zetaPolynomial(Poset)

For the programmer

The object zetaPolynomial is a method function with options.

zetaPolynomial -- computes the zeta polynomial of a poset

Synopsis

Description

See also

Ways to use zetaPolynomial :

For the programmer