Enumeration of Corners in Tree-like Tableaux and a Conjectural (a,b)-analogue

TL;DR

This paper confirms a conjecture on counting corners in tree-like tableaux using bijections with permutations, introduces an ($a$,$b$)-analogue, and discusses implications for the PASEP model.

Contribution

It proves a conjecture on corner enumeration in tree-like tableaux and introduces an ($a$,$b$)-analogue with implications for the PASEP.

Findings

Confirmed the conjecture on corner enumeration in tree-like tableaux.

Established a bijection linking corners to runs of size 1 in permutations.

Introduced an ($a$,$b$)-analogue of the enumeration with PASEP implications.

Abstract

In this paper, we confirm a conjecture of Laborde-Zubieta on the enumeration of corners in tree-like tableaux. Our proof is based on Aval, Boussicault and Nadeau's bijection between tree-like tableaux and permutation tableaux, and Corteel and Nadeau's bijection between permutation tableaux and permutations. This last bijection sends a corner in permutation tableaux to an ascent followed by a descent in permutations, this enables us to enumerate the number of corners in permutation tableaux, and thus to completely solve L.-Z.'s conjecture. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce an ($a$,$b$)-analogue of this enumeration, and explain the implications on the PASEP.