Evilavoiding permutations, introduced by Kim and Williams in 2022, arise in the study of the inhomogeneous totally asymmetric simple exclusion process. Rectangular permutations, introduced by Chirivì, Fang, and Fourier in 2021, arise in the study of Schubert varieties and Demazure modules. Taking a suggestion of Kim and Williams, we supply an explicit bijection between evilavoiding and rectangular permutations in $S_n$ that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a lengthpreserving bijection between words in these regular languages. We extend the bijection to another Wilfequivalent class of permutations, namely the $1$almostincreasing permutations, and exhibit a bijection between rectangular permutations and walks of length $2n2$ in a path of seven vertices starting and ending at the middle vertex.
Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\ldots ,n\}$; the possible arrangements of particles in sites thus correspond to the $n!$ permutations in $S_n$. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on $S_n$, in which two adjacent particles of weights $i<j$ swap places at rate $x_i  y_{n+1j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise, nothing happens.) When $y_i=0$ for all $i$, the stationary distribution was conjecturally linked to Schubert polynomials [18], and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues [4, 5]. In the case of general $y_i$, Cantini [7] showed that $n$ of the $n!$ states have probabilities proportional to double Schubert polynomials. In this paper, we introduce the class of evilavoiding permutations, which are the permutations avoiding the patterns $2413, 4132, 4213,$ and $3214$. We show that there are $\frac {(2+\sqrt {2})^{n1}+(2\sqrt {2})^{n1}}{2}$ evilavoiding permutations in $S_n$, and for each evilavoiding permutation $w$, we give an explicit formula for the steady state probability $\psi _w$ as a product of double Schubert polynomials. (Conjecturally, all other probabilities are proportional to a positive sum of at least two Schubert polynomials.) When $y_i=0$ for all $i$, we give multiline queue formulas for the $\textbf {z}$deformed steady state probabilities and use this to prove the monomial factor conjecture from [18]. Finally, we show that the Schubert polynomials arising in our formulas are flagged Schur functions, and we give a bijection in this case between multiline queues and semistandard Young tableaux.
more » « less NSFPAR ID:
 10407717
 Publisher / Repository:
 Oxford University Press
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 Volume:
 2023
 Issue:
 10
 ISSN:
 10737928
 Page Range / eLocation ID:
 p. 81438211
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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