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 ⌈n/2⌉ into distinct parts. In the second case, an explicit bijection is provided.

Verification and generation of unrefinable partitions

“…It is proved, in particular, that such generators are represented by some unrefinable partitions satisfying conditions on the minimal excludant. Some first combinatorial equalities regarding unrefinable partitions for triangular numbers have been shown recently [ACCL21]. The minimal excludant of a partition, which frequently appears in combinatorial game theory [Gur12,FP15], has also been studied in the context of integers partition by other authors [AN19, BM20, HSS22].…”

“…the least integer that is not a part, which is defined in detail in the next section. According to [ACCL21] we have that , µ = O( √ N ), therefore the algorithm is linear in N in the worst case.…”