Arndt and Carlitz Compositions

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.

💡 Research Summary

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The paper investigates integer compositions under novel adjacency constraints that blend two previously studied families: Carlitz compositions, where consecutive parts must be distinct, and Arndt compositions, which impose an inequality on each ordered pair ((c_{2i-1},c_{2i})). The authors first define the Carlitz‑Arndt compositions (CA(n)) as those compositions of (n) satisfying (c_{2i-1}\neq c_{2i}) for every (i); the last part is unrestricted when the total number of parts is odd. By separating compositions into odd‑length ((CA_o)) and even‑length ((CA_e)) families, they construct explicit bijections:

1. (CA_o(n) \leftrightarrow CA(n-1)) by decreasing the final part (or removing it if it equals 1);
2. (CA_e(n) \leftrightarrow CA(n-2) \cup CA(n-3)) by transforming the last two parts according to whether the penultimate part equals 1.

These bijections imply the recurrence \