On Powers of Some Power Series

TL;DR

This paper explores how compositions of positive integers can be represented using powers of certain power series over any commutative ring, leading to new formulas and connections to recent results.

Contribution

It introduces a novel interpretation of compositions via power series powers and derives new closed formulas for compositions and generalized compositions.

Findings

Derived closed formulas for compositions and generalized compositions.

Connected new formulas to recent research results.

Provided a power series framework applicable over arbitrary rings.

Abstract

We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized compositions with a fixed number of parts are derived. Some results on compositions obtained in some recent papers are consequences of these formulas.