A permutation cycle is a group orbit of the cyclic group generated by a permutation of a finite
set, with the cyclic order induced by that permutation.
Permutation cycles are called "orbits" by Comtet (1974, p. 256). For
example, in the permutation 
, (143) is a 3-cycle and (2) is a 1-cycle. Here, the
notation (143) means that starting from the original ordering 
, the first element is replaced by the fourth, the
fourth by the third, and the third by the first, i.e., 
.
There is a great deal of freedom in picking the representation of a cyclic decomposition since (1) the cycles are disjoint and can therefore be specified in any order, and
(2) any rotation of a given cycle specifies the same cycle (Skiena 1990, p. 20).
Therefore, (431)(2), (314)(2), (143)(2), (2)(431), (2)(314), and (2)(143) all describe
the same permutation. The following table gives the
set of representations for each element of the symmetric
group on three elements, 
, sorted in lowest canonical order (first by cycle length,
and then by lowest initial order of elements).
permutation of  | notation |
 | (1)(2)(3) |
 | (1)(23) |
 | (3)(12) |
 | (123) |
 | (132) |
 | (2)(13) |
The cyclic decomposition of a permutation can be computed in the Wolfram Language with
the function PermutationCycles[p]
and the permutation corresponding to a cyclic decomposition
can be computed with PermutationList[c].
Here, the individual cycles are represented using the function Cycles.
In previous versions, the cyclic decomposition could be computed less efficiently
using ToCycles[p]
in the Wolfram Language package Permutations`
and the permutation corresponding to a cyclic decomposition
could be computed using FromCycles[
c1, ..., cn
] in the Wolfram
Language package Permutations` . According to Vardi (1991), the Wolfram
Language code for ToCycles is one of the most obscure ever written.
Every permutation of 
elements can be expressed as a product
of disjoint cycles, uniquely up to their order and cyclic rotation within each cycle
(Skiena 1990, p. 20). The lengths of the cycles determine the conjugacy
class of the permutation in the symmetric
group 
.
The number 
of permutations of 
elements having exactly 
cycles is given by
 |
(1)
|
where 
are the Stirling numbers of the first
kind. More generally, for a positive integer 
and nonnegative
integers 
and 
,
let 
count permutations of 
elements having exactly 
cycles, each of length at least 
. 
are sometimes called the associated Stirling
numbers of the first kind (Comtet 1974, p. 256). The quantities 
appear in a closed-form expression for the coefficients
in Stirling's series (Comtet 1974, p. 257
and 267). The following table gives the triangles for 
.
 | Sloane |  |
| 1 | A130534 | 1; 1, 1; 2, 3, 1; 6, 11, 6, 1; 24, 50, 35, 10, 1; ... |
| 2 | A008306 | 1; 2; 6, 3; 24, 20; 120, 130, 15; 720, 924, 210; ... |
| 3 | A050211 | 2; 6; 24; 120, 40; 720, 420; 5040, 3948; 40320, ... |
| 4 | A050212 | 6; 24; 120; 720; 5040, 1260; 40320, 18144; ... |
| 5 | A050213 | 24; 120; 720; 5040; 40320; 362880, 72576; ... |
The functions 
satisfy, for 
and 
,
the recurrence relation
 |
(2)
|
where 
is the falling factorial. The initial
conditions are 
and 
for 
,
together with
 |  |  |
(3)
|
 |  |  |
(4)
|
(Riordan 1958, p. 85; Comtet 1974, p. 257).
