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Descent polynomials for permutations with bounded drop size

Chung, Fan and Claesson, Anders and Dukes, Mark and Graham, Ronald (2010) Descent polynomials for permutations with bounded drop size. European Journal of Combinatorics, 31 (7). pp. 1853-1867.

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Abstract

Motivated by juggling sequences and bubble sort, we examine permutations on the set {1,2,…,n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients.