Counting compositions over finite abelian groups
Combinatorics
2017-10-19 v1
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.
Cite
@article{arxiv.1710.06797,
title = {Counting compositions over finite abelian groups},
author = {Zhicheng Gao and Andrew MacFie and Qiang Wang},
journal= {arXiv preprint arXiv:1710.06797},
year = {2017}
}