SOLUTIONS

BUILTIN MATHEMATICA SYMBOL
CycleIndexPolynomial
CycleIndexPolynomial[perm, {x_{1}, ..., x_{n}}]
constructs the cycle index monomial of the permutation perm in the variables .
CycleIndexPolynomial[group, {x_{1}, ..., x_{n}}]
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, {x_{1}, ..., x_{k}}] returns a monic monomial for a permutation perm whose cyclic structure contains 1cycles, 2cycles, etc.
 CycleIndexPolynomial[group, {x_{1}, ..., x_{k}}] returns a polynomial in which the coefficient of the monomial gives the number of group elements whose cyclic structure contains 1cycles, 2cycles, 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.
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