MathDB

A permutation of a finite set is a one-to-one function from the set onto itself. A cycle in a permutation

P

is a nonempty sequence of distinct items

x_1

\ldots\,

x_n

such that

P(x_1) = x_2

P(x_2) = x_3

\ldots\,

P(x_n) = x_1

. Note that we allow the 1-cycle

x_1

where

P(x_1) = x_1

and the 2-cycle

x_1, x_2

where

P(x_1) = x_2

and

P(x_2) = x_1

. Every permutation of a finite set splits the set into a finite number of disjoint cycles. If this number equals 2, then the permutation is called bi-cyclic. Compute the number of bi-cyclic permutations of the 7-element set formed by the letters of "PROBLEM".

Problem Statement