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

Riccardo Aragona, Roberto Civino, Lorenzo Campioni

TL;DR

The paper develops a geometric framework for unrefinable partitions by leveraging the Keith–Nath transformation to connect partitions to numerical sets and their Young diagrams. A central hook-length criterion characterizes unrefinability purely through diagram structure, enabling constructive, non-enumerative proofs of bijections between maximal unrefinable partitions and partitions into distinct parts in both triangular and nontriangular weight regimes. In the triangular case, maximal unrefinable partitions correspond to partitions of $k=(n+1)/2$ into distinct parts, realized via a quasi-symmetric diagram with an extra column; in the nontriangular cases, self-conjugate diagrams or quasi-symmetric patterns yield bijections with odd- and even-distinct partitions, respectively. The results unify and extend previous enumerative classifications, provide a transparent geometric interpretation, and suggest broad applicability to refinement phenomena in partition theory and related combinatorial structures.

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.

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

TL;DR

The paper develops a geometric framework for unrefinable partitions by leveraging the Keith–Nath transformation to connect partitions to numerical sets and their Young diagrams. A central hook-length criterion characterizes unrefinability purely through diagram structure, enabling constructive, non-enumerative proofs of bijections between maximal unrefinable partitions and partitions into distinct parts in both triangular and nontriangular weight regimes. In the triangular case, maximal unrefinable partitions correspond to partitions of into distinct parts, realized via a quasi-symmetric diagram with an extra column; in the nontriangular cases, self-conjugate diagrams or quasi-symmetric patterns yield bijections with odd- and even-distinct partitions, respectively. The results unify and extend previous enumerative classifications, provide a transparent geometric interpretation, and suggest broad applicability to refinement phenomena in partition theory and related combinatorial structures.

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.
Paper Structure (15 sections, 41 theorems, 116 equations, 9 figures, 4 tables)

This paper contains 15 sections, 41 theorems, 116 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Organization of the paper
  4. Preliminaries
  5. Unrefinable partitions and related facts
  6. Numerical sets and Young diagrams
  7. Triangular weight
  8. Proof of $\widetilde{\mathcal{U}}_{T_{n}} \hookrightarrow \mathbb{D}_k$
  9. Proof of $\mathbb{D}_k \hookrightarrow \widetilde{\mathcal{U}}_{T_{n}}$
  10. Proof of unrefinability
  11. Nontriangular weight with maximal part $2n-5$
  12. Proof of $\widetilde{\mathcal{U}}_{T_{n,n-2k}} \hookrightarrow \mathbb{D}^{\,\text{o}}_{2k+2}$
  13. Proof of $\mathbb{D}^{\text{o }}_{2k+2} \hookrightarrow \widetilde{\mathcal{U}}_{T_{n,n-2k}}$
  14. Proof of unrefinability
  15. nontriangular weight with maximal part $2n-4$

Key Result

Lemma 2.6

Let $Y$ be a Young diagram. If $i,j\neq1$, then

Figures (9)

  • Figure 1: The Young diagram corresponding to the numerical set of Example \ref{['ex:ns']}.
  • Figure 2: Symmetry between first row and first column in $Y_{S_\lambda}$, highlighting the special cell $z$.
  • Figure 3: Principal square and first column beyond it in $Y_{S_\lambda}$, illustrating the existence of $z$ cells in the diagonal and in column $C_{z+1}$.
  • Figure 4: Young diagram of a maximal unrefinable partition illustrating the main diagonal, the extra column and its symmetries.
  • Figure 5: Young diagram $Y_{2\eta^*}$ for $\eta=(1,3,4)$.
  • ...and 4 more figures

Theorems & Definitions (99)

  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Definition 2.3
  • Remark 2
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7: tutacs2019young
  • ...and 89 more