Generalized Euler numbers and ordered set partitions

2025/01/13 by Bruce E. Sagan, Sagan, Bruce E. · 2 citations
Mathematics · #11B68 (Primary) 05A18 #11P83 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2501.07692

openalex publication_date 2025/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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+x²/2!+x⁴/4!+...). 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 Möbius inversion over partially ordered sets.

Cited by

• An identity relating Catalan numbers to tangent numbers with arithmetic applications
• Ordered set partition posets

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