# The alternating run polynomials of permutations

## TL;DR

This paper explores the properties of alternating run polynomials in permutations, generalizes related identities, and introduces semi-gamma-positivity, providing new combinatorial interpretations and connections between different permutation classes.

## Contribution

It presents a generalization of the David-Barton identity, offers a combinatorial interpretation of q-alternating run polynomials, and introduces the concept of semi-gamma-positivity for these polynomials.

## Key findings

- Generalized David-Barton identity relating alternating run and Eulerian polynomials
- Combinatorial interpretation of q-alternating run polynomials using grammars
- Proved semi-gamma-positivity of alternating run polynomials of dual Stirling permutations

## Abstract

In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family of q-alternating run polynomials. Furthermore, we introduce the definition of semi-gamma-positive polynomial and we show the semi-gamma-positivity of the alternating run polynomials of dual Stirling permutations. A connection between the up-down run polynomials of permutations and the alternating run polynomials of dual Stirling permutations is established.