On the number of even roots of permutations

Abstract

Let σ be a permutation on n letters. We say that a permutation τ is an even (resp. odd) kth root of σ if τ k = σ and τ is an even (resp. odd) permutation. In this article we obtain generating functions for the number of even and odd kth roots of permutations. Our result implies know generating functions of Moser and Wyman and also some generating functions for sequences in The On-line Encyclopedia of Integer Sequences (OEIS).