A bijection for descent sets of permutations with only even and only odd cycles

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, Heged\H{u}s 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.