A new refinement of Euler numbers on counting alternating permutations

Masato Kobayashi

Published 2019-08-02Version 1

In mathematics, we often encounter surprising interactions with two topics from seemingly different areas. 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.