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  1. Evil-avoiding 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 evil-avoiding and rectangular permutations in $S_n$ that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a length-preserving bijection between words in these regular languages. We extend the bijection to another Wilf-equivalent class of permutations, namely the $1$-almost-increasing permutations, and exhibit a bijection between rectangular permutations and walks of length $2n-2$ in a path of seven vertices starting and ending at the middle vertex.

     
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    Free, publicly-accessible full text available October 6, 2024
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    Abstract We show that, as n goes to infinity, the free group on n generators, modulo {n+u} random relations, converges to a random group that we give explicitly. This random group is a non-abelian version of the random abelian groups that feature in the Cohen–Lenstra heuristics. For each n , these random groups belong to the few relator model in the Gromov model of random groups. 
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