# CycleIndicatorsΒΆ

Enumeration by cycle indices.

alternating: Integer -> SymmetricPolynomial Fraction Integer

`alternating n` is the cycle index of the alternating group of degree `n`.

cap: (SymmetricPolynomial Fraction Integer, SymmetricPolynomial Fraction Integer) -> Fraction Integer

`cap(s1, s2)`, introduced by Redfield, is the scalar product of two cycle indices.

complete: Integer -> SymmetricPolynomial Fraction Integer

`complete n` is the `n` th complete homogeneous symmetric function expressed in terms of power sums. Alternatively it is the cycle index of the symmetric group of degree `n`.

cup: (SymmetricPolynomial Fraction Integer, SymmetricPolynomial Fraction Integer) -> SymmetricPolynomial Fraction Integer

`cup(s1, s2)`, introduced by Redfield, is the scalar product of two cycle indices, in which the power sums are retained to produce a cycle index.

cyclic: Integer -> SymmetricPolynomial Fraction Integer

`cyclic n` is the cycle index of the cyclic group of degree `n`.

dihedral: Integer -> SymmetricPolynomial Fraction Integer

`dihedral n` is the cycle index of the dihedral group of degree `n`.

elementary: Integer -> SymmetricPolynomial Fraction Integer

`elementary n` is the `n` th elementary symmetric function expressed in terms of power sums.

eval: SymmetricPolynomial Fraction Integer -> Fraction Integer

`eval s` is the sum of the coefficients of a cycle index.

graphs: Integer -> SymmetricPolynomial Fraction Integer

`graphs n` is the cycle index of the group induced on the edges of a graph by applying the symmetric function to the `n` nodes.

powerSum: Integer -> SymmetricPolynomial Fraction Integer

`powerSum n` is the `n` th power sum symmetric function.

SFunction: List Integer -> SymmetricPolynomial Fraction Integer

`SFunction(li)` is the `S`-function of the partition `li` expressed in terms of power sum symmetric functions.

skewSFunction: (List Integer, List Integer) -> SymmetricPolynomial Fraction Integer

`skewSFunction(li1, li2)` is the `S`-function of the partition difference `li1 - li2` expressed in terms of power sum symmetric functions.

wreath: (SymmetricPolynomial Fraction Integer, SymmetricPolynomial Fraction Integer) -> SymmetricPolynomial Fraction Integer

`wreath(s1, s2)` is the cycle index of the wreath product of the two groups whose cycle indices are `s1` and `s2`.