Number of factors in K-cycle decompositions of permutations

Abstract

Let π be a permutation in S_n, the symmetric group on n letters. Let k be an integer, 2 ⩽ k ⩽ n, and suppose that π can be represented as a product of k-cycles. Denote by f_k(π) the minimal number of k-cycles required for such a representation. In this paper bounds for f_k(π) are derived. These bounds are used to determine $\mathop {\lim }\limits_{n \to \infty }$ for all k, where f_k(n)=max{f_k(π) | π ε S_n}.