On the classification of the unrefinable partitions into distinct parts

A partition into distinct parts is called “refinable” if one of its parts can be replaced by two different positive integers that are not in the partition, and whose sum equals the replaced part. Otherwise, the partition is “unrefinable.” For example, the partition {1,2,3,5,9,10,12} is refinable because 10 can be replaced by 4 and 6, or 12 by 4 and 8. However, the resulting partitions, such as {1,2,3,4,5,8,9,12}, become unrefinable as no parts can be replaced further. Unrefinable partitions into distinct parts were introduced in previous work of Aragona et al. [“Unrefinable partitions into distinct parts in a normalizer chain”], where they were linked to the generators of a chain of normalizers of the translation group on a suitable Sylow 2-subgroup of the symmetric group on 2^n letters. The logarithm of the ratio between successive normalizers in the chain is independent of n for certain indices and relates to the number of partitions of a given number into at least two distinct parts. Further research connected a specific ratio in the chain to the number of unrefinable partitions of n with a condition on their minimal excluded, which is the smallest positive integer not appearing in the partition. This thesis studies unrefinable partitions into distinct parts, which have not been extensively investigated. The unrefinable condition imposes non-trivial restrictions on the distribution of the parts. We first explored arithmetic properties of unrefinable partitions, leading to the development of two algorithms: one that can identify whether a sequence is an unrefinable partition, and another that enumerates all unrefinable partitions of a given weight. Analysis of these algorithms’ data revealed a significant constraint on the size of the largest part in unrefinable partitions. We established an upper bound for the largest part and defined “maximal” unrefinable partitions as those that reach this bound. We provided a classification of maximal unrefinable partitions for triangular numbers and completed the general classification of maximal unrefinable partitions. These classifications enabled us to establish explicit connections between unrefinable partitions and specific subsets of partitions into distinct parts. For instance, we observed that the number of maximal unrefinable partitions associated with a triangular number can be directly related to the number of distinct-part partitions of a certain size. Specifically, if n=2k-1 the count of maximal unrefinable partitions corresponds to the number of partitions into distinct parts of size k, while if n is even, there is only one such partition. For numbers that are not triangular, the relationship becomes more complex and depends on how close the number is to the next triangular number. In these cases, the count of unrefinable partitions can be described based on whether the difference from the triangular number is even or odd, and whether n is odd or even. Moreover, there is an important connection between unrefinable partitions and numerical semigroups, which are additive submonoids of non-negative integers that include zero and have finitely many elements in their complement. This relationship allowed us to devise additional methods for identifying unrefinable partitions by examining the hooksets of the Young tableau associated with the numerical semigroup. Additionally, we can describe subsets of unrefinable partitions by fixing the largest part and maximizing the number of missing parts, finding a relation with symmetric numerical semigroups when the largest part is a prime number. Future research could focus on understanding this connection more deeply and exploring combinatorial properties of unrefinable partitions to define their generating functions or determine their density.