Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The number of maximal unrefinable partitions

Riccardo Aragona, Lorenzo Campioni, Roberto Civino

TL;DR

The paper advances the theory of maximal unrefinable partitions by completely classifying such partitions for non-triangular integers N=T_{n,d}, linking their counts to partitions into distinct parts with parity constraints that depend on the distance from the next triangular number. It develops a constructive framework centered on the baseline partition π_{n,d}, derives sharp bounds on the largest part λ_t, and uses bijections to polynomials of partitions into distinct parts to obtain exact counting formulas. The principal contributions are the explicit bounds on λ_t across the four target cases (λ_t ∈ {2n-2,2n-3,2n-4,2n-5}), and the closed-form counting results that express the numbers of maximal unrefinable partitions in terms of D_r and D_r^{odd}. Together with detailed constructions and tables, these results complete the classification for all N between consecutive triangular numbers and raise open questions about nonconstructive proofs and potential generalizations beyond maximality.

Abstract

This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer coincides with the number of suitable partitions into distinct parts, depending on the distance from the successive triangular number.

The number of maximal unrefinable partitions

TL;DR

The paper advances the theory of maximal unrefinable partitions by completely classifying such partitions for non-triangular integers N=T_{n,d}, linking their counts to partitions into distinct parts with parity constraints that depend on the distance from the next triangular number. It develops a constructive framework centered on the baseline partition π_{n,d}, derives sharp bounds on the largest part λ_t, and uses bijections to polynomials of partitions into distinct parts to obtain exact counting formulas. The principal contributions are the explicit bounds on λ_t across the four target cases (λ_t ∈ {2n-2,2n-3,2n-4,2n-5}), and the closed-form counting results that express the numbers of maximal unrefinable partitions in terms of D_r and D_r^{odd}. Together with detailed constructions and tables, these results complete the classification for all N between consecutive triangular numbers and raise open questions about nonconstructive proofs and potential generalizations beyond maximality.

Abstract

This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer coincides with the number of suitable partitions into distinct parts, depending on the distance from the successive triangular number.
Paper Structure (13 sections, 25 theorems, 90 equations, 2 figures, 2 tables)

This paper contains 13 sections, 25 theorems, 90 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Related works
  4. Organization of the paper
  5. Construction of maximal unrefinable partitions
  6. The case $\lambda_t=2n-2$
  7. The case $\lambda_t=2n-3$
  8. The case $\lambda_t=2n-4$
  9. The case $\lambda_t=2n-5$
  10. Counting maximal unrefinable partitions
  11. The case $\lambda_t=2n-4$
  12. The case $\lambda_t=2n-5$
  13. Conclusions and open problems

Key Result

Theorem 1

Let $n \in \mathbb N$. The number of maximal unrefinable partitions of $T_n$ is one if $n$ is even and coincides with the number of partitions of $(n+1)/2$ into distinct parts if $n$ is odd.

Figures (2)

  • Figure 1: The three areas of the parts in an unrefinable partition
  • Figure 2: The number of maximal unrefinable partitions between two consecutive triangular numbers. Here $n$ is an even number.

Theorems & Definitions (53)

  • Theorem : aragona2021maximal
  • Definition 1.1
  • Definition 2.1
  • Lemma 2.2: aragona2021maximal
  • proof
  • Proposition 2.3
  • proof
  • Remark 1
  • Remark 2: Anti-symmetry
  • Proposition 2.4
  • ...and 43 more