Descent set distribution for permutations with cycles of only odd or only even lengths
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912221771071488 |
|---|---|
| author | Adin, Ron M. Hegedűs, Pál Roichman, Yuval |
| author_facet | Adin, Ron M. Hegedűs, Pál Roichman, Yuval |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03507 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Descent set distribution for permutations with cycles of only odd or only even lengths Adin, Ron M. Hegedűs, Pál Roichman, Yuval Combinatorics 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. |
| title | Descent set distribution for permutations with cycles of only odd or only even lengths |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.03507 |