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arXiv · 1511.05434

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

Alice L. L. Gao·Emily X. L. Gao·Patxi Laborde-Zubieta·Brian Y. Sun
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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.

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Alice L. L. Gao, Emily X. L. Gao, Patxi Laborde-Zubieta, Brian Y. Sun. 2016-01-06. Enumeration of Corners in Tree-like Tableaux and a Conjectural (a,b)-analogue. https://arxiv.org/abs/1511.05434

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