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

Hongshu Lin, Wenston J. T. Zang

TL;DR

The paper generalizes Euler's partition theorem to $k$-regular partitions by introducing $E_k(n)$ and proving $|B_k(n)|=|E_k(n)|$, extending classical equinumerosity results. It also establishes $|B'_k(n)|=|C_k(n+1)|$ through both a generating-function approach and a separate bijective proof, enriching the landscape of partition identities. The methods yield new combinatorial constructions and $q$-series identities, with $k=2$ recovering familiar Euler-type correspondences while providing novel perspectives on partition bijections and their analytic counterparts.

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 $B_k(n)$.

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

TL;DR

The paper generalizes Euler's partition theorem to -regular partitions by introducing and proving , extending classical equinumerosity results. It also establishes through both a generating-function approach and a separate bijective proof, enriching the landscape of partition identities. The methods yield new combinatorial constructions and -series identities, with recovering familiar Euler-type correspondences while providing novel perspectives on partition bijections and their analytic counterparts.

Abstract

Let denote the set of -distinct partitions of , and let be the set of -regular partitions of . Glaisher showed that . For , this equality yields the celebrated Euler's partition theorem. In this paper, we present a new partition set , which is equinumerous to .

Paper Structure

This paper contains 3 sections, 5 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm-4']}
  3. Proof of Theorem \ref{['thm-2']}

Key Result

Theorem 1.1

For $n\geq 0$, we have where $C(n)$ is the set of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the set of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

Theorems & Definitions (5)

  • Theorem 1.1: Andrews-2025
  • Theorem 1.2: Glaisher-1883
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5