An insertion algorithm over staircase tableaux compatible with the ASEP's matrix ansatz

TL;DR

This paper introduces a new insertion algorithm for staircase tableaux that provides a combinatorial proof for the ASEP's stationary probabilities, revealing recursive structures and extending to type B symmetric tableaux.

Contribution

It presents a novel insertion algorithm over staircase tableaux that aligns with the ASEP matrix ansatz and extends to type B symmetric tableaux, offering new combinatorial insights.

Findings

Recursive structure of staircase tableaux derived from the insertion algorithm

Factorised formulas for generating polynomials of staircase tableaux

Bijection with coloured inversion tables

Abstract

Based on the matrix ansatz of Derrida, Evans, Hakim and Pasquier, we prensent a new way of computing the stationary probability of a state of the asym- metric simple exclusion process (ASEP). Through an insertion algorithm over staircase tableaux, we give a combinatorial proof to the current interpretation of the ASEP by these tableaux of Corteel and Williams. The insertion algorithm induces a recursive structure which implies nice factorised formulas for the generating polynomials of staircase tableaux, as well as a bijection with some coloured inversion tables. In addi- tion, we adapt the insertion algorithm to the case of type B symmetric tableaux and we define a new matrix ansatz compatible with it.