Descent polynomials for permutations with bounded drop size

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