A Geometric Characterization of Maximal Unrefinable Partitions via the Keith-Nath Transformation and Young Diagrams

TL;DR

This paper offers a geometric criterion based on hook lengths in Young diagrams to characterize unrefinable partitions, providing new combinatorial proofs and insights into their structural properties.

Contribution

It introduces a geometric approach to characterize unrefinable partitions via Young diagrams and hook lengths, extending previous enumerative results with combinatorial proofs.

Findings

Hook length criterion characterizes unrefinable partitions.

Young diagrams are quasi-symmetric in the triangular case.

Diagrams are self-conjugate or quasi-symmetric in the nontriangular case.

Abstract

We investigate the combinatorial structure of unrefinable partitions through their correspondence with numerical sets and Young diagrams. Building on the bijection introduced by Keith and Nath, we apply a general geometric criterion that links the unrefinability of a partition directly to the hook lengths of its associated Young diagram. This criterion provides a structural method for the characterization of any unrefinable partition. Using this general framework, we revisit the correspondence results between maximal unrefinable partitions and partitions into distinct parts, previously established using enumerative methods. We provide alternative and purely combinatorial proofs of these bijections, focusing on the rigid symmetry structures of the Young diagrams. In the triangular weight case, we show that the corresponding diagrams are quasi-symmetric, i.e. symmetric up to a single extra column. We extend this analysis to the nontriangular case, showing that the diagrams either exhibit this same quasi-symmetric structure or are perfectly self-conjugate, depending on the maximal part.