characteristicPolynomial -- computes the characteristic polynomial of a ranked poset with a unique minimal element

Synopsis

Usage:

p = characteristicPolynomial P

p = characteristicPolynomial(P, VariableName => symbol)
Inputs:
- P, an instance of the type Poset, a ranked poset
Optional inputs:
- VariableName => a symbol, default value q
Outputs:
- p, a ring element, the characteristic polynomial of $P$

Description

The characteristic polynomial of a ranked poset is the generating function with variable $q$ such that the coefficient of $q^r$ is the sum overall vertices of rank $r$ of the Moebius function of $v$.

The characteristic polynomial of the chain of $n$ is $q^{n-1}(q-1)$.

i1 : n = 5;

i2 : factor characteristicPolynomial chain n

        3
o2 = (q) (q - 1)

o2 : Expression of class Product

And the characteristic polynomial of the booleanLattice of $n$ is $(q-1)^n$.

i3 : factor characteristicPolynomial booleanLattice n

            5
o3 = (q - 1)

o3 : Expression of class Product

Ways to use characteristicPolynomial :

characteristicPolynomial(Poset)

For the programmer

The object characteristicPolynomial is a method function with options.

characteristicPolynomial -- computes the characteristic polynomial of a ranked poset with a unique minimal element

Synopsis

Description

See also

Ways to use characteristicPolynomial :

For the programmer