Cyclic descents for near-hook and two-row shapes

Ron M. Adin, Sergi Elizalde, Yuval Roichman

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

A notion of cyclic descents on standard Young tableaux (SYT) of rectangular shape was introduced by Rhoades, and extended to certain skew shapes by the last two authors. The cyclic descent set restricts to the usual descent set when the largest value is ignored, and has the property that the number of SYT of a given shape with a given cyclic descent set D is invariant under cyclic shifts of the entries of D. Following these results, the existence of cyclic descent sets for standard Young tableaux of any skew shape other than a ribbon was conjectured by the authors, and recently proved by Adin, Reiner and Roichman. Unfortunately, the proof does not provide a natural definition of the cyclic descent set for a specific tableau. In this paper we explicitly describe cyclic descent sets for SYT of (possibly skew) shapes which either have exactly two rows or are near-hooks, i.e., are one cell away from a hook. Our definition provides a constructive combinatorial proof of the existence of cyclic descent sets for these shapes, and coincides with that of Rhoades for two-row rectangular shapes. We also show that cyclic descent sets for near-hook shaped tableaux are unique.

Original languageEnglish
Pages (from-to)152-178
Number of pages27
JournalEuropean Journal of Combinatorics
Volume79
DOIs
StatePublished - Jun 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Ltd

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