A bijection between evil-avoiding and rectangular permutations

TL;DR

This paper constructs an explicit bijection between evil-avoiding and rectangular permutations, linking their combinatorial structures and extending to related permutation classes and lattice walks.

Contribution

It provides the first explicit bijection between evil-avoiding and rectangular permutations, preserving key permutation statistics and connecting these classes through regular languages and lattice paths.

Findings

Established a length-preserving bijection between evil-avoiding and rectangular permutations.

Extended the bijection to 1-almost-increasing permutations, showing their Wilf-equivalence.

Connected rectangular permutations to walks in a seven-vertex path, revealing structural insights.

Abstract

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 Chiriv\`i, 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.