A new combinatorial formula for alternating descent polynomials

TL;DR

This paper introduces a new combinatorial formula for alternating descent polynomials of types A and B, providing a unified combinatorial proof connecting these polynomials with Hoffman's derivative polynomials for tangent and secant functions.

Contribution

It presents a novel combinatorial formula for alternating descent polynomials and unifies their connection with Hoffman's derivative polynomials through a combinatorial proof.

Findings

New combinatorial formula for alternating descent polynomials

Unified proof linking these polynomials with Hoffman's derivative polynomials

Enhanced understanding of the combinatorial structure of these polynomials

Abstract

We prove a combinatorial formula for the alternating descent polynomials of type A and B. Combining with Josuat-Verg\`es' combinatorial interpretation for Hoffman's derivative polynomials for tangent and secant functions, we obtain a unified combinatorial proof of Ma-Yeh's two formulae linking these two families of polynomials.