Generalized Euler numbers and ordered set partitions

TL;DR

This paper introduces a combinatorial model for generalized Euler numbers based on ordered set partitions, providing new proofs and properties using combinatorial techniques like involutions and Möbius inversion.

Contribution

It presents a novel combinatorial interpretation of generalized Euler numbers and applies combinatorial methods to establish their properties and relations.

Findings

Established a recursion for generalized Euler numbers

Proved integrality and congruence properties

Developed combinatorial proofs using involutions and Möbius inversion

Abstract

The Euler numbers have been widely studied. A signed version of the Euler numbers of even subscript are given by the coefficients of the exponential generating function 1/(1+x^2/2!+x^4/4!+...). Leeming and MacLeod introduced a generalization of the Euler numbers depending on an integer parameter d where one takes the coefficients of the expansion of 1/(1+x^d/d!+x^{2d}/(2d)!+...). These numbers have been shown to have many interesting properties despite being much less studied. And the techniques used have been mainly algebraic. We propose a combinatorial model for them as signed sums over ordered partitions. We show that this approach can be used to prove a number of old and new results including a recursion, integrality, and various congruences. Our methods include sign-reversing involutions and M\"obius inversion over partially ordered sets.