Occupied Corners in Tree-like Tableaux

Patxi Laborde-Zubieta

Séminaire lotharingien de combinatoire, Tome 74 (2015-2018)

Résumé

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.