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

TL;DR

This paper refines the known equality between permutations with odd or even cycle lengths by showing a descent set distribution equivalence, using generating functions and higher Lie character identities.

Contribution

It introduces a descent set refinement of the cycle length parity symmetry in permutations, employing new identities on higher Lie characters.

Findings

Number of permutations with a given descent set and odd cycles equals those with the complementary set and even cycles.

Extension of the result to permutations in $S_{2n+1}$.

Use of generating functions and higher Lie character identities in proof.

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.