A bijection for descent sets of permutations with only even and only odd cycles
Sergi Elizalde
TL;DR
The paper delivers a fully combinatorial proof of the refined Descent/Ascents equality for permutations with restricted ascent/descent sets by translating permutations into multisets of primitive necklaces via classical bijections and then into Lyndon-factorized words. A new weight-preserving bijection $\Psi$ (with inverse $\Omega$) between words with odd distinct Lyndon factors and words with all even Lyndon factors completes the construction, while generating-function arguments provide a concise alternative for the Lyndon-factor identity. The resulting $f_S=\Phi_S^{-1}\circ\Psi\circ\Xi_S$ is an explicit bijection between the relevant permutation classes, yielding the desired equality for any $S\subseteq[n-1]$. This work clarifies the combinatorial structure behind the refinement and links it to classic necklace/Lyndon word theory, offering practical tools for related descent/ascents problems and potential connections to higher Lie characters. The approach also highlights a compact generating-function route for the Lyndon-factor result and situates Foata-type transformations within the broader necklace-Lyndon framework.
Abstract
It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Hegedűs and Roichman recently found a surprising refinement of this identity. They showed that, for any fixed set $J$, the equality still holds when restricting to permutations with descent set $J$ on one side, and permutations with ascent set $J$ on the other. Their proof uses generating functions for higher Lie characters, and it also yields a version for odd $n$. Here we give a bijective proof of their result. We first use known bijections, due to Gessel, Reutenauer and others, to restate the identity in terms of multisets of necklaces, which we interpret as words, and then describe a new weight-preserving bijection between words all of whose Lyndon factors have odd length and are distinct, and words all of whose Lyndon factors have even length. We also show that the corresponding equality about Lyndon factorizations has a short proof using generating functions.