Cyclic Sieving of Increasing Tableaux and small Schr\"oder Paths

TL;DR

This paper establishes a bijection between 2-row increasing tableaux and small Schr"oder paths, explores their relation to jeu de taquin and tropical frieze patterns, and demonstrates new cyclic sieving phenomena.

Contribution

It introduces a novel bijection linking increasing tableaux and Schr"oder paths, and extends cyclic sieving results to new combinatorial structures.

Findings

Bijection between 2-row increasing tableaux and small Schr"oder paths

Connections between jeu de taquin and tropical frieze patterns

New instances of cyclic sieving phenomenon for these structures

Abstract

An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schr\"oder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, and D. White, generalizing results of D. White and of J. Stembridge.