Distribution of the Number of Corners in Tree--like Tableaux

TL;DR

This paper investigates the distribution of corners in tree-like tableaux, revealing that their number follows a normal distribution asymptotically, and connects these combinatorial structures to particle system models like PASEP.

Contribution

It establishes the asymptotic normality of the number of corners in tree-like tableaux and explores their connection to the PASEP model.

Findings

Number of corners is asymptotically normally distributed.

Tree-like tableaux are connected to permutation tableaux and PASEP.

The total number of corners relates to the system's current activity.

Abstract

In this paper, we study tree--like tableaux and some of their probabilistic properties. Tree--like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of an interacting particles system. In particular, in the context of tree-like tableaux, a corner corresponds to a node occupied by a particle that could jump to the right while inner corners indicate a particle with an empty node to its left. Thus, the total number of corners represents the number of nodes at which PASEP can move, i. e. the total current activity of the system. As the number of inner corners and regular corners is connected, we limit our discussion to just regular corners and show that asymptotically, the number of corners in a tableau of length $n$ is normally distributed.