Partitions of an Integer into Powers

TL;DR

This paper investigates the structure and enumeration of integer partitions into powers of a fixed integer, introducing new structural insights and efficient enumeration methods through lattice and tree frameworks.

Contribution

It extends known results on partitions into powers, establishing a distributive lattice structure and providing a new tree-based enumeration method.

Findings

Partitions form a distributive lattice structure.

A tree structure enables efficient enumeration.

Generalizes previous enumeration results.

Abstract

In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.