On the maximal part in unrefinable partitions of triangular numbers

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.