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arXiv · 1505.06098

Occupied corners in tree-like tableaux

Patxi Laborde Zubieta
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Abstract

Tree-like tableaux are combinatorial objects that appear in a combinatorial understanding of the PASEP model from statistical mechanics. In this understanding, the corners of the Southeast border correspond to the locations where a particle may jump to the right. Such corners may be of two types: either empty or occupied. Our main result is the following: on average there is one occupied corner per tree-like tableau. We give two proofs of this, a short one which gives us a polynomial version of the result, and another one using a bijection between tree-like tableaux and permutations which gives us an additional information. Moreover, we obtain the same result for symmetric tree-like tableaux and we refine our main result to an equivalence class. Finally we present a conjecture enumerating corners, and we explain its consequences for the PASEP.

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Patxi Laborde Zubieta. 2015-05-22. Occupied corners in tree-like tableaux. https://arxiv.org/abs/1505.06098

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