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).

• Type: Journal
• Year: 2016
• Where: EJC
• Authors: Joanna N. Chen, William Y. C. Chen
• Keywords: EJC 2016, Information Technology