Inversion-descent polynomials for restricted permutations

Abstract

We derive generating functions for a variety of distributions of joint permutation statistics all of which involve a bound on the maximum drop size of a permutation π, i.e., max{i−π(i)}. Our main result treats the case for the joint distribution of the number of inversions, the number of descents and the maximum drop size of permutations on [n]={1,2,…,n}. A special case of this (ignoring the number of inversions) connects with earlier work of Claesson, Dukes and the authors on descent polynomials for permutations with bounded drop size. In that paper, the desired numbers of permutations were given by sampling the coefficients of certain polynomials Qk. We find a natural interpretation of all the coefficients of the Qk in terms of a restricted version of Eulerian numbers.