Cyclic descents for Motzkin paths

TL;DR

This paper introduces explicit cyclic descent sets for Motzkin paths, demonstrating their equidistribution with those of three-row standard Young tableaux and providing a bijective proof of their shuffling property.

Contribution

It explicitly describes cyclic descent sets for Motzkin paths and proves their equidistribution with three-row SYT, addressing a previously open problem.

Findings

Cyclic descent sets for Motzkin paths are explicitly characterized.

The cyclic descent sets for Motzkin paths are shown to be equidistributed with those of three-row SYT.

A bijective proof of the shuffling property for Motzkin paths is provided.

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 Adin, Elizalde and Roichman. 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. Adin, Reiner and Roichman proved that a skew shape has a cyclic descent map if and only if it is not a connected ribbon. Unfortunately, their proof is nonconstructive. Recently Huang constructed an explicit cyclic descent map for all shapes where this is possible. In the earlier version of Adin, Elizalde and Roichman's paper, they asked to find statistics on combinatorial objects which are equidistributed with cyclic descents on SYT of given shapes. In this paper, we explicitly describe cyclic descent sets for Motzkin paths, which are equidistributed with cyclic descent sets of SYT for three-row shapes. Moreover, in light of Stanley's shuffling theorem, we give a bijective proof of the shuffling property of descent statistics for Motzkin paths.