Derivative polynomials and enumeration of permutations by their alternating descents

TL;DR

This paper derives an explicit formula for counting permutations based on their alternating descents and investigates the zero distribution of their generating polynomials.

Contribution

It introduces a new explicit formula for permutation counts with alternating descents and analyzes the zeros of their generating polynomials.

Findings

Explicit formula for permutation enumeration by alternating descents

Interlacing property of zeros of generating polynomials

Insights into the roots' real parts distribution

Abstract

In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. Moreover, we study the interlacing property of the real parts of the zeros of the generating polynomials of these numbers.