A new refinement of Euler numbers on counting alternating permutations

TL;DR

This paper introduces a novel refinement of Euler numbers to better understand their combinatorial properties related to alternating permutations, building on previous work by Heneghan and Petersen.

Contribution

It provides a new refinement of Euler numbers that addresses specific combinatorial relations involving alternating permutations.

Findings

New refinement of Euler numbers introduced

Addresses specific combinatorial relations

Enhances understanding of alternating permutations

Abstract

At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer the combinatorial question on some particular relation of Euler numbers proved by Heneghan-Petersen, Power series for up-down min-max permutations, College Math. Journal, Vol. 45, No. 2 (2014), 83-91.