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.
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