CycleIndicatorsΒΆ

cycles.spad line 1

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.