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

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...

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Published in:arXiv.org 2016-01
Main Authors: Gao, Alice L L, Gao, Emily X L, Laborde-Zubieta, Patxi, Sun, Brian Y
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Language:English
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Ascent
Corners
Enumeration
Permutations
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description 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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subjects Ascent
Corners
Enumeration
Permutations
title Enumeration of Corners in Tree-like Tableaux and a Conjectural (a,b)-analogue
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