The Relationship Between Euler Numbers and Bernoulli Numbers with Ordered Partitions

TL;DR

This paper establishes recursive formulas and combinatorial interpretations linking functions related to ordered partitions to Bernoulli and Euler numbers, revealing new structural insights and explicit relationships.

Contribution

It introduces recursive definitions and combinatorial formulas for functions connected to Bernoulli and Euler numbers, expanding understanding of their combinatorial and algebraic properties.

Findings

Derived explicit formulas relating Bernoulli and Euler numbers to functions K_b and K_e.

Provided recursive definitions for K_b and K_e functions.

Presented combinatorial interpretations of K_b and K_e via ordered partitions.

Abstract

In this paper, for every $n \in \mathbb{N}$, the following relationships between the functions $K_{b}(n)$ and $K_{e}(n)$ and the Bernoulli and Euler numbers are proved: \[ B_{2n} = -\,\frac{(2n)!}{2^{2n}-2}\, K_{b}(n), \qquad E_{2n} = (2n)!\, K_{e}(n). \] The functions $K_{b}$ and $K_{e}$ are defined recursively by \[ K_{b}(0) = K_{e}(0) = 1, \] \[ K_{b}(n) = - \sum_{n'=0}^{\,n-1} \frac{K_{b}(n')}{\bigl( 2(n-n') + 1 \bigr)!}, \qquad n \ge 1, \] \[ K_{e}(n) = - \sum_{n'=0}^{\,n-1} \frac{K_{e}(n')}{\bigl( 2(n-n') \bigr)!}, \qquad n \ge 1. \] Furthermore, we present combinatorial interpretations of these functions in terms of ordered partitions of $n$: \[ K_{b}(n) = \sum_{\lambda \vDash n} \frac{(-1)^{\ell(\lambda)}} {\displaystyle\prod_{i=1}^{\ell(\lambda)} (2b_i + 1)!}, \qquad n \ge 1, \] \[ K_{e}(n) = \sum_{\lambda \vDash n} \frac{(-1)^{\ell(\lambda)}} {\displaystyle\prod_{i=1}^{\ell(\lambda)} (2b_i)!}, \qquad n \ge 1, \] where $\lambda = (b_1,b_2,\ldots,b_k) \vDash n$ and $\ell(\lambda)=k$.