# CycleIndexPolynomial

CycleIndexPolynomial[perm,{x1,,xn}]

constructs the cycle index monomial of the permutation perm in the variables xi.

CycleIndexPolynomial[group,{x1,,xn}]

constructs the cycle index polynomial of group in the variables xi.

# Details • CycleIndexPolynomial[perm,vars] assumes perm is a permutation acting on the domain {1,,max}, where max is the largest integer moved, as given by PermutationMax[perm].
• CycleIndexPolynomial[perm,vars,n] denotes that perm acts on a domain of n points, which must be equal to or larger than the largest moved point.
• CycleIndexPolynomial[perm,{x1,,xk}] returns a monic monomial x1a1x2a2 xkak for a permutation perm whose cyclic structure contains a1 1-cycles, a2 2-cycles, etc.
• CycleIndexPolynomial[group,{x1,,xk}] returns a polynomial in which the coefficient of the monomial x1a1x2a2 xkak gives the number of group elements whose cyclic structure contains a1 1-cycles, a2 2-cycles, etc., divided by the order of the group. It is the average of the cycle index monomials of its elements.
• Variables corresponding to cycle lengths not present in the elements of the group are ignored.
• If the elements of the group contain cycle lengths beyond the number of variables provided, then the result effectively uses a value 1 for those missing variables.
• The length of the cycles of a permutation or a permutation group is always bounded above by the length of their support, as given by PermutationLength. Hence, this is a safe estimate for the number of variables to include as the second argument of CycleIndexPolynomial.

# Examples

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## Basic Examples(2)

Cycle index monomial of a permutation:

 In:= Out= Cycle index polynomial for the alternating group on five points:

 In:= Out= ## Properties & Relations(4)

Introduced in 2012
(9.0)