Interpolating Classical Partitions of the Set of Positive Integers

TL;DR

This paper introduces a family of integer partitions that interpolate between classical Beatty partitions and 2-adic partitions, analyzing their structure and interactions, revealing new insights into partition relationships.

Contribution

It constructs a unified family of partitions of positive integers that smoothly transition between known partition types and studies their interrelations.

Findings

Partition family interpolates between Beatty and 2-adic partitions

Detailed analysis of interactions between two Beatty partitions

Restrictions and partial results on interactions involving the extended partition

Abstract

We construct an easily described family of partitions of the positive integers into $n$ disjoint sets with essentially the same structure for every $n \geq 2$. In a special case, it interpolates between the Beatty $\frac{1}{\phi} + \frac{1}{\phi^2} = 1$ partitioning ($n=2$) and the 2-adic partitioning in the limit as $n \rightarrow \infty$. We then analyze how membership of elements in the sets of one partition relates to membership in the sets of another. We investigate in detail the interactions of two Beatty partitions with one another and the interactions of the $\phi$ Beatty partition mentioned above with its "extension" to three sets given by the construction detailed in the first part. In the first case, we obtain detailed results whereas the second case we place some restrictions on the interaction but cannot obtain exhaustive results.