Combinatorial proofs of two Euler type identities due to Andrews

TL;DR

This paper provides combinatorial proofs of identities related to Euler's partition theorem, establishing equalities among various partition counting functions using bijections and decorated partitions.

Contribution

It introduces new combinatorial bijections to prove identities previously established by generating functions, expanding understanding of partition identities.

Findings

Proved combinatorially that a(n)=b(n) and b(n)=c(n).

Established that c_1(n)=b_1(n) through combinatorial arguments.

Extended identities to cases with parts occurring twice or thrice.

Abstract

We prove combinatorially some identities related to Euler's partition identity (the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts). They were conjectured by Beck and proved by Andrews via generating functions. Let $a(n)$ be the number of partitions of $n$ such that the set of even parts has exactly one element, $b(n)$ be the difference between the number of parts in all odd partitions of $n$ and the number of parts in all distinct partitions of $n$, and $c(n)$ be the number of partitions of $n$ in which exactly one part is repeated. Then, $a(n)=b(n)=c(n)$. The identity $a(n)=c(n)$ was proved combinatorially (in greater generality) by Fu and Tang. We prove combinatorially that $a(n)=b(n)$ and $b(n)=c(n)$. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. Let $c_1(n)$ be the number of partitions of $n$ such that there is exactly one part occurring three times while all other parts occur only once and let $b_1(n)$ to be the difference between the total number of parts in the partitions of $n$ into distinct parts and the total number of different parts in the partitions of $n$ into odd parts. We prove combinatorially that $c_1(n)=b_1(n)$. In addition to these results by Andrews, we prove combinatorially that $b_1(n)=a_1(n)$, where $a_1(n)$ counts partitions of $n$ such that the set of even parts has exactly one element and satisfying some additional conditions. We also treat the case when exactly one part occurs twice while all other parts occur only once.