Enriched Cycle Structures and Roots of Permutations
William Y. C. Chen, Elena L. Wang
TL;DR
This work develops a purely combinatorial framework that links $r$-regular and $r$-cycle permutations through $r$-enriched structures, culminating in an explicit bijection ${\rm Reg}_r(rn)\leftrightarrow {\rm Cyc}^*_r(rn)$. Using these tools, it derives a strengthened BMW-type inequality and provides a combinatorial foundation for when a random permutation has an $r$-th root, first for primes and then for prime powers. A key result is the monotonicity of $p_r(n)$ for $r=q^l$, established via extended criterion and lemmas, with precise equality conditions. Collectively, the paper unifies existing results (BMW, Knopfmacher–Warlimont, Chernoff) within a cohesive combinatorial approach and deepens understanding of the relationship between cycle structure and permutation roots.
Abstract
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 $\{1,2,\ldots, n\}$ 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 $r\geq 2$, we define an $r$-enriched permutation as a permutation with $r$-singular cycles colored by one of the colors $1, 2, \ldots, r-1$. 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 $n+1 \not\equiv 0 \pmod q$, the equality $p_r(n)=p_r(n+1)$ has been established by Chernoff.