CycleIndicators¶

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.