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

TL;DR

This work refines the classical descent-set equidistribution between permutations with odd-cycle-only and even-cycle-only structures by incorporating descent sets, proving that for $S_{2n}$ the number of permutations with Descent set $J$ and all cycles odd equals the number with complementary Descent set and all cycles even (with an analogous odd-size variant). The authors develop a robust generating-function framework for higher Lie characters, expressing intricate sums over cycle types as explicit exponentials involving Möbius functions, and connect these to quasisymmetric function identities via the GR correspondence. They also relate these character-theoretic results to root enumerators, establishing both odd-root enumerator equalities and signed variants, and show that the odd-root enumerator is induced from a type-$B$ subgroup, providing a unifying view via induction from $B_n$. Overall, the paper links descent-set combinatorics, permutation cycle structure, and representation theory to produce precise equidistribution results with substantial structural insight and broad consequences for symmetric-group character theory.

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.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Lemma 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4