What question did this study set out to answer?

The goal is to refine the known equality of permutations with odd and even cycle lengths by involving descent sets.

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March 5, 2026Open Access

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

Q: What does this research mean for the field?

The number of permutations in the symmetric group 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. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Key Points

The goal is to refine the known equality of permutations with odd and even cycle lengths by involving descent sets.
Analyzed permutations in the symmetric group with odd and even cycle lengths.
Established a new equality using descent sets and complementary cycles.
Utilized generating functions for character values and introduced a new identity for higher Lie characters.
Demonstrated that the number of permutations with a prescribed descent set and odd cycles equals those with a complementary set and even cycles.
Outlined a variant for permutations in an extended symmetric group configuration.

Abstract

It is known that the number of permutations in the symmetric group S 2 n 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 2 n 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 2 n + 1 . The proof uses generating functions for character values and applies a new identity on higher Lie characters.

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

Key Points

Abstract

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