Counting on Euler and Bernoulli Number Identities

TL;DR

This paper introduces combinatorial proofs for two identities involving Euler and Bernoulli numbers, providing an alternative to traditional analytical or inductive methods.

Contribution

It presents novel combinatorial proofs for Euler and Bernoulli number identities using up-down permutations, expanding the methods used in this area.

Findings

Two new identities proved combinatorially

Alternative proofs to traditional analytical methods

Enhanced understanding of Euler and Bernoulli number relationships

Abstract

While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down permutations.