Abstract
The problem of finding Schur-positive sets of permutations, originally posed by Gessel and Reutenauer, has seen some recent developments. Schur-positive sets of pattern-avoiding permutations have been found by Sagan et al. and a general construction based on geometric operations on grid classes has been given by the authors. In this paper we prove that horizontal rotations of Schur-positive subsets of permutations are always Schur-positive. The proof applies a cyclic action on standard Young tableaux of certain skew shapes and a jeu-de-taquin type straightening algorithm. As a consequence of the proof we obtain a notion of cyclic descent set on these tableaux, which is rotated by the cyclic action on them.
| Original language | English |
|---|---|
| Pages (from-to) | 121-137 |
| Number of pages | 17 |
| Journal | Journal of Combinatorial Theory. Series A |
| Volume | 152 |
| DOIs | |
| State | Published - Nov 2017 |
Bibliographical note
Publisher Copyright:© 2017 Elsevier Inc.
Funding
We thank Ron Adin for useful discussions and two anonymous referees for helpful suggestions and comments. The first author was partially supported by Simons Foundation grant #280575 and NSA grant H98230-14-1-0125. The second author was partially supported by MISTI MIT-Israel Seed Fund and by Dartmouth's Shapiro visitors fund.
| Funders | Funder number |
|---|---|
| MISTI MIT-Israel Seed Fund | |
| Simons Foundation | 280575 |
| National Security Agency | H98230-14-1-0125 |
Keywords
- Cyclic action
- Cyclic descent
- Horizontal rotation
- Schur-positivity
- Standard Young tableau