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Mathematics > Combinatorics

Title: Combinatorial proofs of two Euler type identities due to Andrews

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. Moreover, we offer an analog of these results for the number of partitions of $n$ with exactly one part occurring two times while all other parts occur only once.
Comments: 14 pages
Subjects: Combinatorics (math.CO); Number Theory (math.NT)
MSC classes: 05A17, 11P81
Cite as: arXiv:1803.06394 [math.CO]
  (or arXiv:1803.06394v1 [math.CO] for this version)

Submission history

From: Cristina Ballantine [view email]
[v1] Fri, 16 Mar 2018 20:44:46 GMT (10kb)