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arXiv · 1710.06797

Counting compositions over finite abelian groups

Zhicheng Gao·Andrew MacFie·Qiang Wang
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Abstract

We find the number of compositions over finite abelian groups under two types of restrictions: (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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Zhicheng Gao, Andrew MacFie, Qiang Wang. 2017-10-18. Counting compositions over finite abelian groups. https://arxiv.org/abs/1710.06797

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