S-restricted compositions revisited

Behrouz Zolfaghari, M. S. Fallah*, Mehdi Sedighi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number9
JournalDiscrete Mathematics and Theoretical Computer Science
Volume19
Issue number1
Publication statusPublished - 2017
Externally publishedYes

Keywords

  • Closed-form formula
  • Diophantine equations
  • Homogeneous recurrence relations
  • Restricted compositions

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)
  • Discrete Mathematics and Combinatorics

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