Descent set distribution for permutations with cycles of only odd or only even lengths

• DOI: 10.48550/arxiv.2502.03507
• Publication Date: 2025-01-01
• Authors: Adin, Ron M.; Hegedűs, Pál; Roichman, Yuval
• Keywords: FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO)

It is known that the number of permutations in the symmetric group $S_{2n}$ with cycles of odd lengths only is equal to the number of permutations with cycles of even lengths only. We prove a refinement of this equality, involving descent sets: the number of permutations in $S_{2n}$ with a prescribed descent set and all cycles of odd lengths is equal to the number of permutations with the complementary descent set and all cycles of even lengths. There is also a variant for $S_{2n+1}$. The proof uses generating functions for character values and applies a new identity on higher Lie characters.<br/>23 pages. arXiv admin note: substantial text overlap with arXiv:2312.08904<br/>