# Wolfram Language & System 10.0 (2014)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# CycleIndexPolynomial

CycleIndexPolynomial[perm,{x1,,xn}]
constructs the cycle index monomial of the permutation perm in the variables .

CycleIndexPolynomial[group,{x1,,xn}]
constructs the cycle index polynomial of group in the variables .

## DetailsDetails

• CycleIndexPolynomial[perm,vars] assumes perm is a permutation acting on the domain , 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 for a permutation perm whose cyclic structure contains 1-cycles, 2-cycles, etc.
• CycleIndexPolynomial[group,{x1,,xk}] returns a polynomial in which the coefficient of the monomial gives the number of group elements whose cyclic structure contains 1-cycles, 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.

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Cycle index monomial of a permutation:

 Out[1]=

Cycle index polynomial for the alternating group on five points:

 Out[1]=