# CycleIndicatorsΒΆ

cycles.spad line 1 [edit on github]

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`

.