Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.14508 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912967294976000 |
|---|---|
| author | Shankar, Umesh |
| author_facet | Shankar, Umesh |
| contents | A conjecture by Deutsch, Kitaev, and Remmel states that the triples of permutation statistics $(S_{10}, S_{12}, S_{17})$ and $(S_{12}, S_{10} ,S_{17})$ are equidistributed over the symmetric group $\mathfrak{S}_n$. Here, $S_{10}$ enumerates descents with odd descent tops, $S_{12}$ enumerates odd-odd adjacent pairs, and $S_{17}$ records the largest integer $i$ such that $1, 2, \dots, i$ appear in left-to-right order. In this note, we resolve this conjecture affirmatively by providing a bijective proof. We introduce an insertion process that constructs a recursive involution on $\mathfrak{S}_n$ that swaps $S_{10}$ and $S_{12}$ while keeping $S_{17}$ unchanged. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14508 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An insertion process and a parity based equidistribution Shankar, Umesh Combinatorics 05A05, 05A19 A conjecture by Deutsch, Kitaev, and Remmel states that the triples of permutation statistics $(S_{10}, S_{12}, S_{17})$ and $(S_{12}, S_{10} ,S_{17})$ are equidistributed over the symmetric group $\mathfrak{S}_n$. Here, $S_{10}$ enumerates descents with odd descent tops, $S_{12}$ enumerates odd-odd adjacent pairs, and $S_{17}$ records the largest integer $i$ such that $1, 2, \dots, i$ appear in left-to-right order. In this note, we resolve this conjecture affirmatively by providing a bijective proof. We introduce an insertion process that constructs a recursive involution on $\mathfrak{S}_n$ that swaps $S_{10}$ and $S_{12}$ while keeping $S_{17}$ unchanged. |
| title | An insertion process and a parity based equidistribution |
| topic | Combinatorics 05A05, 05A19 |
| url | https://arxiv.org/abs/2603.14508 |