Let denote the number of cycles of length for a permutation expressed as a product of disjoint cycles. The cycle index of a permutation group of order and degree is then the polynomial in variables , , ..., given by the formula
(1)
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The cycle index of a permutation group is implemented as CycleIndexPolynomial[perm, x1, ..., xn], which returns a polynomial in . For any permutation , the numbers satisfy
(2)
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and thus constitutes a partition of the integer . Sets of values are commonly denoted , where ranges over all the -vectors satisfying equation (2).
Formulas for the most important permutation groups (the symmetric group , alternating group , cyclic group , dihedral group , and trivial group ) are given by
(3)
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(4)
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(5)
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(6)
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(7)
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where means divides and is the totient function (Harary 1994, p. 184).