Compositions of Integers With Bounded Parts

TL;DR

This paper studies ordered partitions of integers with bounded parts, providing methods to construct, count, and generate functions for such compositions, advancing understanding of constrained integer compositions.

Contribution

It introduces a new method for constructing all bounded compositions and derives explicit formulas and generating functions for their enumeration.

Findings

Provided a construction method for bounded compositions.

Derived explicit formulas for counting compositions.

Developed generating functions for compositions with bounded parts.

Abstract

In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function describing the number of $k$-tuples whose entries are bounded in this way and sum to a fixed value $g$.