Statistics on permutations with bounded drop size

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.