Derivative polynomials and permutations by numbers of interior peaks and left peaks

TL;DR

This paper explores the relationship between derivative polynomials derived from tangent and secant functions and the enumeration of interior and left peaks in permutations, providing new insights into their generating functions and properties.

Contribution

It establishes a novel connection between derivative polynomials and permutation peak statistics, including recurrence relations and real-rootedness of their generating functions.

Findings

Coefficients of derivative polynomials relate to interior and left peaks.

Generated functions exhibit recurrence relations and real-rootedness.

Provides combinatorial interpretations of derivative polynomial coefficients.

Abstract

Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left peaks over the symmetric group. Properties of the generating functions for the numbers of interior and left peaks over the symmetric group, including recurrence relations, generating functions and real-rootedness, are studied.