Rhombic alternative tableaux and assembl\'ees of permutations

TL;DR

This paper introduces rhombic alternative tableaux and their connection to assemblées of permutations, providing combinatorial formulas for the steady state probabilities of a two-species ASEP model.

Contribution

It establishes a bijection between rhombic alternative tableaux and assemblées of permutations, and provides an insertion algorithm for their weight generating functions.

Findings

Enumerates tableaux using Lah numbers.

Provides a bijective proof linking tableaux and assemblées.

Derives the partition function for the two-species ASEP at q=1.

Abstract

In this paper, we introduce the rhombic alternative tableaux, whose weight generating functions provide combinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there are two species of particles, one heavy and one light, hopping right and left on a one-dimensional finite lattice with open boundaries. Parameters $\alpha$, $\beta$, and $q$ describe the hopping probabilities. The rhombic alternative tableaux are enumerated by the Lah numbers, which also enumerate certain assembl\'ees of permutations. We describe a bijection between the rhombic alternative tableaux and these assembl\'ees. We also provide an insertion algorithm that gives a weight generating function for the assembl\'ees. Combined, these results give a bijective proof for the weight generating function for the rhombic alternative tableaux, which is also the partition function of the two-species ASEP at $q=1$.