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arXiv · 2502.03507

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

Ron M. Adin·Pál Hegedűs·Yuval Roichman
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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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Ron M. Adin, Pál Hegedűs, Yuval Roichman. 2025-02-05. Descent set distribution for permutations with cycles of only odd or only even lengths. https://arxiv.org/abs/2502.03507

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