Statistics on permutations with bounded drop size

TL;DR

This paper studies permutations with limited drop size, deriving generating functions for various permutation statistics and demonstrating their equidistribution, thus advancing understanding of their combinatorial properties.

Contribution

It extends the analysis of bounded permutations by providing generating functions for key statistics and establishing their equidistribution, using Petersen's method.

Findings

Generated functions for (inv, lmax) and (DIS, cyc) over bounded permutations

Proved equidistribution of certain permutation statistics

Derived generating function of des over 213-avoiding bounded permutations

Abstract

Permutations with bounded drop size, which we also call bounded permutations, was introduced by Chung, Claesson, Dukes and Graham. Petersen introduced a new Mahonian statistic the sorting index, which is denoted by $\sor$. Meanwhile, Wilson introduced the statistic $\DIS$, which turns out to satisfy that $\sor(\sigma)=\DIS(\sigma^{-1})$ for any permutation $\sigma$. In this paper, we maintain Petersen's method to deduce the generating functions of $(\inv, \lmax)$ and $(\DIS, \cyc)$ over bounded permutations to show their equidistribution. Moreover, the generating function of $\des$ over $213$-avoiding bounded permutations and some related equidistributions are given as well.