Enriched Cycle Structures and Roots of Permutations

This paper is concerned with a duality between $r$-regular permutations and $r$-cycle permutations, and a monotone property due to Bóna-McLennan-White on the probability $p_r(n)$ for a random permutation of $\{1,2,\dots ,n\}$ to have an $r$-th root, where $r$ is a prime. For $r=2$, the duality relates permutations with odd cycles to permutations with even cycles. To handle the general case where $r\ge 2$, we define an $r$-enriched permutation as a permutation with $r$-singular cycles colored by one of the colors $1,2,\dots ,r-1$. In this setup, we discover a bijection between $r$-regular permutations and enriched $r$-cycle permutations, which in turn yields a stronger version of an inequality of Bóna-McLennan-White. This leads to a fully combinatorial understanding of the monotone property, thereby answering their question. When $r$ is a prime power $q^l$, we further show that $p_r(n)$ is monotone. In the case that $n+1\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}q)$, the equality $p_r(n)=p_r(n+1)$ has been established by Chernoff.