Cycles and fixed points

In mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S. These orbits are subsets of S that can be written as { c1, ..., cn }, such that The corresponding cycle of π is written as ( c1 c2 ... cn ); this expression is not unique since c1 can be chosen to be any element of the orbit.

The size n of the orbit is called the length of the corresponding cycle; when n = 1, the single element in the orbit is called a fixed point of the permutation.

It is typical, but not necessary, to not write the cycles of length one in such an expression.

[2][3] The value f(k, j) counts the number of permutations of k elements with exactly j fixed points.

(3) There are three different methods to construct a permutation of k elements with j fixed points: