We find the number of compositions over finite abelian groups under two types of restriction: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring’s problem over finite fields can be converted to type (i) compositions, whereas Carlitz and “locally Mullen” compositions can be formulated as type (ii) compositions. We use the multisection formula to translate the problem from integers to group elements, the transfer matrix method to do exact counting, and finally the Perron-Frobenius theorem to derive asymptotics. We also exhibit bijections involving certain restricted classes of compositions.

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Keywords Enumeration, Finite abelian group, Integer composition, Transfer matrix
Journal Electronic Journal of Combinatorics
Gao, Z. (Zhicheng), MacFie, A. (Andrew), & Wang, Q. (2018). Counting compositions over finite abelian groups. Electronic Journal of Combinatorics, 25(2).