Descent polynomials for permutations with bounded drop size

Fan Chung,Mark Dukes,Anders Claesson,Ronald Graham

doi:10.46298/dmtcs.2856

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. Motivés par les "suites de jonglerie'' et le tri à bulles, nous étudions les permutations de l'ensemble ${1, 2, \ldots, n}$ ayant $d$ descentes et une taille de déficience maximale $k$. Nous donnons des formules explicites pour l'énumération de telles permutations pour des entiers k et d fixés, ainsi que les fonctions génératrices connexes. Nous montrons aussi que les coefficients possèdent des propriétés d'unimodalité et de symétrie.

Highlights

There have been extensive studies of various statistics on Sn, the set of all permutations of {1, 2, . . . , n}
For a permutation π in Sn, we say that π has a drop at i if πi < i and that the drop size is i − πi
One of the earliest results [8] in permutation statistics states that the number of permutations in Sn with k drops equals the number of permutations with k descents

Summary

Introduction

There have been extensive studies of various statistics on Sn, the set of all permutations of {1, 2, . . . , n}. Given a permutation π in Sn, let Des(π) = {1 ≤ i < n : πi > πi+1} be the descent set of π and let des(π) = | Des(π)| be the number of descents. Let bn,k(r) = |{π ∈ Bn,k : des(π) = r}|, and define the (k-maxdrop-restricted) descent polynomial

Results

Conclusion