Descent polynomials for permutations with bounded drop size

TL;DR

This paper provides explicit formulas and generating functions for counting permutations with a fixed number of descents and bounded drop size, revealing their unimodality and symmetry properties.

Contribution

It introduces new explicit formulas and generating functions for permutations with bounded drop size and descents, advancing combinatorial enumeration methods.

Findings

Explicit formulas for permutations with bounded drop size and descents

Derived generating functions for these permutations

Proved unimodality and symmetry of the coefficients

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.