Abstract
In this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux.
Highlights
To motivate our introduction of the rhombic alternative tableaux, we first describe the asymmetric simple exclusion process (ASEP)
The ASEP is a model from statistical physics that describes the dynamics of interacting particles hopping left and right on a one-dimensional finite lattice with open boundaries
The classical ASEP with three parameters is defined by the following hopping probabilities: particles may enter at the left of the lattice with rate α, they may exit at the right with rate β, and in the bulk the probability of hopping left is q times that of hopping right
Summary
To motivate our introduction of the rhombic alternative tableaux, we first describe the asymmetric simple exclusion process (ASEP). [4, 9] revealed a fascinating connection between a two-species generalization of the ASEP with the same five parameters and moments of Koornwinder-Macdonald polynomials, an important class of multi-variate orthogonal polynomials that generalizes the Askey-Wilson polynomials. This two-species ASEP (studied in [2, 11, 16], among others), has two species of particles, heavy and light, with the heavy particles able to enter and exit at the boundaries.
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