Arndt and Carlitz Compositions
Brian Hopkins, Aram Tangboonduangjit

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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Algebraic structures and combinatorial models
