Positivity of Narayana polynomials and Eulerian polynomials

Abstract

Gamma-positive polynomials frequently appear in finite geometries, algebraic combinatorics and number theory. Sagan and Tirrell (2020) [34] stumbled upon some unimodal sequences, which turn out to be alternating gamma-positive instead of gamma-positive. Motivated by this work, we first show that one can derive alternatingly γ-positive polynomials from γ-positive polynomials. We then prove the alternating γ-positivity and Hurwitz stability of several polynomials associated with the Narayana polynomials of types A and B. In particular, by introducing the definition of colored $2\times n$ Young diagrams, we provide combinatorial interpretations for three identities related to the Narayana numbers of type B. Finally, we present several identities involving the Eulerian polynomials of types A and B.