English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We prove combinatorially some identities related to Euler's partitionidentity (the number of partitions of $n$ into distinct parts equals the numberof partitions of $n$ into odd parts). They were conjectured by Beck and provedby Andrews via generating functions.  Let $a(n)$ be the number of partitions of $n$ such that the set of even partshas exactly one element, $b(n)$ be the difference between the number of partsin all odd partitions of $n$ and the number of parts in all distinct partitionsof $n$, and $c(n)$ be the number of partitions of $n$ in which exactly one partis repeated. Then, $a(n)=b(n)=c(n)$. The identity $a(n)=c(n)$ was provedcombinatorially (in greater generality) by Fu and Tang. We provecombinatorially that $a(n)=b(n)$ and $b(n)=c(n)$. Our proof relies onbijections between a set and a multiset, where the partitions in the multisetare 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 partsoccur only once and let $b_1(n)$ to be the difference between the total numberof parts in the partitions of $n$ into distinct parts and the total number ofdifferent parts in the partitions of $n$ into odd parts. We provecombinatorially 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 partitionsof $n$ such that the set of even parts has exactly one element and satisfyingsome additional conditions. We also treat the case when exactly one part occurstwice while all other parts occur only once.

Combinatorial proofs of two Euler type identities due to Andrews