Arndt and Carlitz Compositions

TL;DR

This paper generalizes and enumerates restricted integer compositions, combining previous notions and applying combinatorial and generating function techniques, motivated by gap-free compositions and Rogers-Ramanujan partitions.

Contribution

It introduces a unified framework for restricted compositions, extending prior concepts and providing enumeration results through combinatorial proofs and generating functions.

Findings

Established enumeration formulas for generalized compositions

Connected compositions to Rogers-Ramanujan partitions

Provided combinatorial and generating function proofs

Abstract

Carlitz considered integer compositions in which adjacent parts must be unequal. Arndt recently initiated the study of restricted compositions based on conditions applied to certain pairs of parts rather than to individual parts. Here, we combine and generalize these notions, establishing enumeration results using both combinatorial proofs and generating functions. Motivations for our generalizations include the gap-free compositions studied by Hitczenko and Knopfmacher and the Rogers-Ramanujan integer partitions.