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Séminaire Lotharingien de Combinatoire, B74b (2015), 14 pp.

# Patxi Laborde-Zubieta

# Occupied Corners in Tree-like Tableaux

**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
result, 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 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 concerning the enumeration of corners, and we explain its
consequences for the PASEP.

Received: August 24, 2015.
Revised: February 2, 2016.
Accepted: February 2, 2016.

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