On the maximal part in unrefinable partitions of triangular numbers

TL;DR

This paper investigates unrefinable partitions of triangular numbers, establishing bounds on the largest part, classifying maximal cases, and providing explicit bijections for odd cases, thereby advancing understanding of partition refinement limitations.

Contribution

It provides a new upper bound for the largest part in unrefinable partitions and fully classifies maximal unrefinable partitions of triangular numbers, including explicit bijections.

Findings

Upper bound of O(n^{1/2}) for the largest part in unrefinable partitions

Unique maximal unrefinable partition for even triangular numbers

Number of maximal unrefinable partitions for odd triangular numbers equals partitions of eil(n/2) into distinct parts

Abstract

A partition into distinct parts is refinable if one of its parts $a$ can be replaced by two different integers which do not belong to the partition and whose sum is $a$, and it is unrefinable otherwise. Clearly, the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of 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 complete classification of maximal unrefinable partitions for triangular numbers, proving that if $n$ is even there exists only one maximal unrefinable partition of $n(n+1)/2$, and that if $n$ is odd the number of such partitions equals the number of partitions of $\lceil n/2\rceil$ into distinct parts. In the second case, an explicit bijection is provided.