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Séminaire Lotharingien de Combinatoire, 85B.28 (2021), 12 pp.

# Donghyun Kim and Lauren Williams

# Schubert Polynomials and the Inhomogeneous TASEP on a Ring

**Abstract.**
Consider a lattice of n sites arranged
around a ring, with the *n* sites occupied by
particles of weights {1,2,...,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 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 (1/2) (2+2^{1/2})^{n-1} +
(2-2^{1/2})^{n-1})
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.

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

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