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

2025/02/05 by Ron M. Adin, Adin, Ron M., Pál Hegedüs +3
Computer Science · Decision Sciences · Mathematics · #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Probability and Risk Models #Statistical Distribution Estimation and Applications

paper · pdf · doi:10.48550/arxiv.2502.03507

openalex publication_date 2025/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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