The number of maximal unrefinable partitions

TL;DR

This paper fully classifies maximal unrefinable partitions, linking their count to specific partitions into distinct parts based on the distance from triangular numbers, extending previous work.

Contribution

It completes the classification of maximal unrefinable partitions, generalizing prior results limited to triangular numbers.

Findings

Number of maximal unrefinable partitions equals the count of certain partitions into distinct parts.

The classification depends on the distance from the number to the nearest triangular number.

Extends previous work from triangular numbers to a broader class of integers.

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.