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On Euler's Theorem

George E. Andrews, Rahul Kumar, Ae Ja Yee

TL;DR

The paper generalizes Euler's partition theorem by introducing two refined counting functions, $C(n)$ and $D(n)$, and proving that for $n>0$ the equalities $A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1)$ hold, where $A(n)$ is the number of partitions into distinct parts and $B(n)$ the number into odd parts. It establishes these equalities via both analytic and bijective methods: generating-function proofs show $B(n)=C(n+1)$ and $C(n)=\frac{1}{2}D(n)$, while bijective proofs provide direct combinatorial mappings, including a Glaisher-bijection-based construction for $B(n)=C(n+1)$ and a decrement-based mapping proving $A(n-1)=\frac{1}{2}D(n)$. The results unify and extend Euler’s theorem by linking distinct-part and odd-part partitions to a pair of auxiliary partition statistics, $C(n)$ and $D(n)$. The approaches enrich the landscape of partition identities and offer concrete combinatorial interpretations and tools for related identities.

Abstract

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*} A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1), \end{align*} where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

On Euler's Theorem

TL;DR

The paper generalizes Euler's partition theorem by introducing two refined counting functions, and , and proving that for the equalities hold, where is the number of partitions into distinct parts and the number into odd parts. It establishes these equalities via both analytic and bijective methods: generating-function proofs show and , while bijective proofs provide direct combinatorial mappings, including a Glaisher-bijection-based construction for and a decrement-based mapping proving . The results unify and extend Euler’s theorem by linking distinct-part and odd-part partitions to a pair of auxiliary partition statistics, and . The approaches enrich the landscape of partition identities and offer concrete combinatorial interpretations and tools for related identities.

Abstract

Euler's theorem asserts that where is the number of partitions of into distinct parts and is the number of partitions of into odd parts. In this paper, it is proved that for , \begin{align*} A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1), \end{align*} where is the number of partitions of with largest part even and parts not exceeding half of the largest part are distinct, and is the number of partitions of into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

Paper Structure

This paper contains 6 sections, 1 theorem, 19 equations.

Table of Contents

  1. Introduction
  2. Proof that $B(n)=C(n+1)$
  3. Proof that $C(n)=\frac{1}{2}D(n)$
  4. A Bijective proof of Theorem \ref{['main theorem']}
  5. A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$
  6. A Bijective proof of $B(n)=C(n+1)$

Key Result

Theorem 1.1

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

Theorems & Definitions (2)

  • Theorem 1.1
  • remark 1