S-Restricted Compositions Revisited

TL;DR

This paper develops a formal method to derive closed-form formulas for counting S-restricted compositions of integers, by reducing the problem to solving linear recurrence relations and Diophantine equations, with focus on cases with few coefficients.

Contribution

It introduces a novel formalism linking S-restricted compositions to interpreters and Diophantine equations, enabling explicit formulas for specific cases.

Findings

Derived closed-form formulas for certain S-restricted compositions.

Reduced composition counting to solving linear recurrence relations.

Connected composition problems to solvable Diophantine equations.

Abstract

An S-restricted composition of a positive integer n is an ordered partition of n where each summand is drawn from a given subset S of positive integers. There are various problems regarding such compositions which have received attention in recent years. This paper is an attempt at finding a closed- form formula for the number of S-restricted compositions of n. To do so, we reduce the problem to finding solutions to corresponding so-called interpreters which are linear homogeneous recurrence relations with constant coefficients. Then, we reduce interpreters to Diophantine equations. Such equations are not in general solvable. Thus, we restrict our attention to those S-restricted composition problems whose interpreters have a small number of coefficients, thereby leading to solvable Diophantine equations. The formalism developed is then used to study the integer sequences related to some well-known cases of the S-restricted composition problem.