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2016

On permutations with bounded drop size

3 years 6 months ago
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).
Joanna N. Chen, William Y. C. Chen
Added 02 Apr 2016
Updated 02 Apr 2016
Type Journal
Year 2016
Where EJC
Authors Joanna N. Chen, William Y. C. Chen
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