On the number of even roots of permutations

Abstract

Let $\sigma$ be a permutation on $n$ letters. We say that a permutation $\tau$ is an even (resp. odd) $k$th root of $\sigma$ if ${\tau }^k=\sigma$ and $\tau$ is an even (resp. odd) permutation. In this article, we obtain generating functions for the number of even and odd $k$th 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).

Second version, 11 pages

References in corpus (3)

• A note on the number of $k$-roots in $S_n$
• On the number of mth roots of permutations
• Enumerating kth Roots in the Symmetric Inverse Monoid