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Generalized Euler numbers and ordered set partitions

Bruce E. Sagan

TL;DR

The paper generalizes Euler numbers to $E_n^{(d)}$ via a reciprocal exponential generating function and provides a combinatorial interpretation as signed sums over $d$-divisible ordered partitions. It develops a toolkit based on sign-reversing involutions and Möbius inversion on posets to derive a recursion, integrality, and congruences for $E_n^{(d)}$ and to connect to $d$-alternating permutations. It proves key identities such as $E_{dn}^{(d)} = (-1)^n ext{#} A_{dn}^{(d)}$ and establishes congruences, including $E_{pn}^{(p)} ext{ is congruent to } (-1)^n$ modulo $p^2$ for primes $p$. It outlines future directions, including combinatorial proofs of determinant formulas and connections to Bernoulli-type numbers.

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 Möbius inversion over partially ordered sets.

Generalized Euler numbers and ordered set partitions

TL;DR

The paper generalizes Euler numbers to via a reciprocal exponential generating function and provides a combinatorial interpretation as signed sums over -divisible ordered partitions. It develops a toolkit based on sign-reversing involutions and Möbius inversion on posets to derive a recursion, integrality, and congruences for and to connect to -alternating permutations. It proves key identities such as and establishes congruences, including modulo for primes . It outlines future directions, including combinatorial proofs of determinant formulas and connections to Bernoulli-type numbers.

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 Möbius inversion over partially ordered sets.
Paper Structure (5 sections, 10 theorems, 59 equations, 2 figures, 3 tables)

This paper contains 5 sections, 10 theorems, 59 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The original Euler numbers
  3. Basic properties of generalized Euler numbers
  4. Congruences for generalized Euler numbers
  5. Future work

Key Result

Theorem 1.1

For all $n\ge0$ and $d\ge2$ we have

Figures (2)

  • Figure 1: The signed set of all $2$-divisible partitions of $[4]$ and a sign-reversing involution
  • Figure 2: The subgroup lattice ${\cal L}$ for $C_p\times C_p$

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 6 more