  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.
