On permutations with bounded drop size

The maximum drop size of a permutation π of [n]={1,2,…,n} is defined to be the maximum value of i−π(i). Chung, Claesson, Dukes and Graham found polynomials Pk(x) that can be used to determine the number of permutations of [n] with d descents and maximum drop size at most k. Furthermore, Chung and Graham gave combinatorial interpretations of the coefficients of Qk(x)=xkPk(x) and Rn,k(x)=Qk(x)(1+x+⋯+xk)n−k, and raised the question of finding a bijective proof of the symmetry property of Rn,k(x). In this paper, we construct a map φk on the set of permutations with maximum drop size at most k. We show that φk is an involution and it induces a bijection in answer to the question of Chung and Graham. The second result of this paper is a proof of a unimodality conjecture of Hyatt concerning the type B analogue of the polynomials Pk(x).