On $\gamma$-vectors and the derivatives of the tangent and secant functions

TL;DR

This paper explores the connection between gamma-vectors of Coxeter complexes and associahedra with derivative polynomials of tangent and secant functions, providing grammatical descriptions for these vectors and classical polynomials.

Contribution

It introduces a novel link between gamma-vectors and derivative polynomials, along with grammatical frameworks for these mathematical objects.

Findings

Gamma-vectors derived from tangent and secant derivative polynomials.

Grammatical descriptions for gamma-vectors and classical polynomials.

Connections established between combinatorial structures and analytical functions.

Abstract

In this paper we consider the gamma-vectors of the types A and B Coxeter complexes as well as the gamma-vectors of the types A and B associahedrons. We show that these gamma-vectors can be obtained by using derivative polynomials of the tangent and secant functions. A grammatical description for these gamma-vectors is discussed. Moreover, we also present a grammatical description for the well known Legendre polynomials and Chebyshev polynomials of both kinds.