Arndt and Carlitz Compositions

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.