Authors: Fan Chung ¹; Anders Claesson ²; Mark Dukes ^3,4; Ronald Graham ¹

• 1 University of California [San Diego] [UC San Diego]
• 2 The Mathematics Institute, Reyjavik University
• 3 Science Institute, University of Iceland
• 4 University of Iceland [Reykjavik]

Motivated by juggling sequences and bubble sort, we examine permutations on the set${1, 2, \ldots, n}$ 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.