Journal of Integer Sequences, Vol. 27 (2024), Article 24.3.6

Compositions with an Odd Number of Parts, and Other Congruences

Joshua P. Bowman
Natural Science Division
Seaver College
Pepperdine University
24255 Pacific Coast Highway
Malibu, CA 90263


This article begins with a description of three notions of compositions that have parts in a fixed set of positive integers: linear compositions, circular compositions, and cyclic compositions. We describe some relations among these notions and review generating functions to count each type of composition. The main result of the paper generalizes these known results to count compositions in which the number of parts is required to be congruent to q modulo m for some fixed 0 ≤ qm − 1. The particular case m = 2, q = 1 yields the compositions with an odd number of parts. The latter sections apply the main theorem to several special cases, including compositions in which the parts are allowed to be drawn from a multiset.

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(Concerned with sequences A000004 A000007 A000016 A000032 A000045 A000204 A000225 A001350 A001519 A002878 A004146 A008965 A011782 A025192 A032198 A056295 A088305 A093040 A094686 A100886 A324969 A365857 A365858 A365859 A366043 A366044 A366045.)

Received October 2 2023; revised versions received October 9 2023; February 29 2024; March 1 2024. Published in Journal of Integer Sequences, March 9 2024.

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