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) |
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) |
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
where 
means 
divides 
and 
is the totient function (Harary 1994, p. 184).
See also
Alternating Group, Cycle Graph, Cycle Polynomial, Permutation Cycle, Cyclic Group, Dihedral Group, Permutation Group, Pólya Enumeration Theorem, Simple Graph, Symmetric Group, Symmetric Polynomial
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References
Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 181 and 184, 1994.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Cycle Index." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CycleIndex.html