Rhombic alternative tableaux, assemblees of permutations, and the ASEP

Olya Mandelshtam (1) , Xavier Viennot (2)

1 Department of Mathematics [Berkeley]
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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.

Domaines: Combinatoire [math.CO]

Olya Mandelshtam, Xavier Viennot. Rhombic alternative tableaux, assemblees of permutations, and the ASEP. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6320⟩. ⟨hal-02168122⟩