Generalizations of Euler's Theorem to $k$-regular partitions

TL;DR

This paper introduces a new partition set $E_k(n)$ that is equinumerous with $k$-regular partitions, extending Euler's theorem to broader classes of partitions.

Contribution

It presents a novel partition set $E_k(n)$ that generalizes Euler's theorem to $k$-regular partitions, establishing new combinatorial equivalences.

Findings

$E_k(n)$ is equinumerous with $B_k(n)$ for all $k$

Extends classical Euler's theorem to new partition classes

Provides combinatorial proofs for the new identities

Abstract

Let $A_k(n)$ denote the set of $k$-distinct partitions of $n$, and let $B_k(n)$ be the set of $k$-regular partitions of $n$. Glaisher showed that $\# A_k(n) = \# B_k(n)$. For $k=2$, this equality yields the celebrated Euler's partition theorem. In this paper, we present a new partition set $E_k(n)$, which is equinumerous to $B_k(n)$.