Positivity of Narayana polynomials and Eulerian polynomials

TL;DR

This paper explores the relationships between gamma-positivity and alternating gamma-positivity, deriving new properties of Narayana and Eulerian polynomials, and providing combinatorial interpretations and conjectures in algebraic combinatorics.

Contribution

It establishes that gamma-positive polynomials are also alternatingly semi-gamma-positive and connects these concepts through new identities and interpretations involving Narayana and Eulerian polynomials.

Findings

Proves alternating gamma-positivity of certain polynomial combinations.

Provides combinatorial interpretations using Young diagrams.

Presents conjectures on Boros-Moll polynomials and permutation enumerators.

Abstract

Gamma-positivity appears frequently in finite geometries, combinatorics and number theory. Motivated by the recent work of Sagan and Tirrell (Adv. Math., 374 (2020), 107387), we study the relationships between gamma-positivity and alternating gamma-positivity. As applications, we derive several alternatingly gamma-positive polynomials related to Narayana polynomials and Eulerian polynomials. In particular, we show the alternating gamma-positivity and Hurwitz stability of a combination of the modified Narayana polynomials of types A and B. By using colored $2\times n$ Young diagrams, we present a unified combinatorial interpretations of three identities involving Narayana numbers of type B. A general result of this paper is that every gamma-positive polynomial is also alternatingly semi-gamma-positive. At the end of this paper, we pose two conjectures, one concerns the Boros-Moll polynomials and the other concerns the enumerators of permutations by descents and excedances.