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Descent polynomials for permutations with bounded drop size

Abstract : 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.
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https://hal.inria.fr/hal-01186284
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Fan Chung, Anders Claesson, Mark Dukes, Ronald Graham. Descent polynomials for permutations with bounded drop size. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.247-258. ⟨hal-01186284⟩

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