Cyclic descents for near-hook and two-row shapes

TL;DR

This paper explicitly describes cyclic descent sets for standard Young tableaux of two-row and near-hook shapes, providing a constructive proof of their existence and uniqueness, extending previous theoretical results.

Contribution

The authors give explicit combinatorial descriptions of cyclic descent sets for two-row and near-hook shapes, confirming their existence and uniqueness with constructive methods.

Findings

Explicit cyclic descent sets for two-row shapes

Explicit cyclic descent sets for near-hook shapes

Constructive proof of existence and uniqueness

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 and resulting generating functions 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.