Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations

Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\dots,n\}$; the possible arrangements of particles in sites thus corresponds to the $n!$ permutations in $S_n$. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which two adjacent particles of weights $i<j$ swap places at rate $x_i - y_{n+1-j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise nothing happens.) In the case that $y_i=0$ for all $i$, the stationary distribution was conjecturally linked to Schubert polynomials by Lam-Williams, and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues by Ayyer-Linusson and Arita-Mallick. In the case of general $y_i$, Cantini showed that $n$ of the $n!$ states have probabilities proportional to products of double Schubert polynomials. In this paper we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns $2413, 4132, 4213$ and $3214$. We show that there are $\frac{(2+\sqrt{2})^{n-1}+(2-\sqrt{2})^{n-1}}{2}$ evil-avoiding permutations in $S_n$, and for each evil-avoiding permutation $w$, we give an explicit formula for the steady state probability $ψ_w$ as a product of double Schubert polynomials. We also show that the Schubert polynomials that arise in these formulas are flagged Schur functions, and give a bijection in this case between multiline queues and semistandard Young tableaux.

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