Corners in tree-like tableaux

TL;DR

This paper investigates the number of corners in tree-like tableaux, proving a conjecture about their total count and establishing connections with permutation tableaux, thereby advancing combinatorial understanding of these structures.

Contribution

It proves a conjecture on the total number of corners in tree-like tableaux and links these results to permutation tableaux using bijections.

Findings

Confirmed the conjecture on corners in tree-like tableaux

Established bijections with permutation tableaux

Extended results to symmetric tableaux

Abstract

In this paper, we study tree--like tableaux, combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree--like tableaux and the total number of corners in symmetric tree--like tableaux. In this paper, we prove the first conjecture leaving the proof of the second conjecture to the full version of this paper. Our proofs are based on the bijection with permutation tableaux or type--B permutation tableaux and consequently, we also prove results for these tableaux.