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Posets :: rankGeneratingFunction

rankGeneratingFunction -- computes the rank generating function of a ranked poset

Synopsis

Usage:

r = rankGeneratingFunction P

r = rankGeneratingFunction(P, VariableName => symbol)
Inputs:
- P, an instance of the type Poset, a ranked poset
Optional inputs:
- VariableName => a symbol, default value q
Outputs:
- r, a ring element, the rank generating function of $P$

Description

The rank generating function of $P$ is the polynomial with the coefficient of $q^i$ given by the number of vertices in rank $i$ of $P$.

The rank generating function of the $n$ chain is $q^{n-1} + \cdots + q + 1$.

i1 : n = 5;

i2 : rankGeneratingFunction chain n

      4    3    2
o2 = q  + q  + q  + q + 1

o2 : ZZ[q]

The rank generating function of the $n$ booleanLattice is $(q+1)^n$.

i3 : factor rankGeneratingFunction booleanLattice n

            5
o3 = (q + 1)

o3 : Expression of class Product

See also

isRanked -- determines if a poset is ranked
rankPoset -- generates a list of lists representing the ranks of a ranked poset

Ways to use `rankGeneratingFunction` :

"rankGeneratingFunction(Poset)"

For the programmer

The object rankGeneratingFunction is a method function with options.