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/

Global: Zendy Plus annual

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

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A partition into distinct parts is refinable if one of its parts $a$ can bereplaced by two different integers which do not belong to the partition andwhose sum is $a$, and it is unrefinable otherwise. Clearly, the condition ofbeing unrefinable imposes on the partition a non-trivial limitation on the sizeof the largest part and on the possible distributions of the parts. We prove a$O(n^{1/2})$-upper bound for the largest part in an unrefinable partition of$n$, and we call maximal those which reach the bound. We show a completeclassification of maximal unrefinable partitions for triangular numbers,proving that if $n$ is even there exists only one maximal unrefinable partitionof $n(n+1)/2$, and that if $n$ is odd the number of such partitions equals thenumber of partitions of $\lceil n/2\rceil$ into distinct parts. In the secondcase, an explicit bijection is provided.

On the maximal part in unrefinable partitions of triangular numbers