Séminaire Lotharingien de Combinatoire, 95B.5 (2026), 12 pp.
Ron M. Adin, Sergi Elizalde, Pál Hegedüs, and Yuval Roichman
Descent Sets of Permutations with Only Even or Only Odd Cycle Lengths
Abstract.
It is known that the number of permutations in the symmetric group
S2n consisting of cycles of odd length is equal to the number of permutations consisting of cycles of even length.
We prove a surprising refinement of this equality: the number of permutations in S2n with a prescribed ascent set J and all cycles of odd length is equal to the number of permutations with descent set J and all cycles of even length.
We present two different proofs of this result, as well as a similar
statement for S2n+1. The first proof is
algebraic, and it involves a new identity on generating functions for
higher Lie characters. The second proof is bijective, and it
introduces a new bijection between words with distinct Lyndon factors
of odd length and words with even Lyndon factors.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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