On permutations with bounded drop size

The maximum drop size of a permutation it of [n] = {1, 2, ... , n} is defined to be the maximum value of i - pi(i). Chung, Claesson, Dukes and Graham found polynomials P-k(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 Q(k)(x) = x(k)P(k)(x) and R-n,R-k(x) = Q(k)(X) (1 + x + ... + x(k))(n-k), and raised the question of finding a bijective proof of the symmetry property of R-n,R-k(x). In this paper, we construct a map phi(k) on the set of permutations with maximum drop size at most k. We show that phi(k) 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 P-k(x). (C) 2015 Elsevier Ltd. All rights reserved.