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.
Zhicheng Gao; Andrew MacFie; Qiang Wang. Counting compositions over finite abelian groups. The electronic journal of combinatorics, Tome 25 (2018) no. 2. doi: 10.37236/7591
@article{10_37236_7591,
author = {Zhicheng Gao and Andrew MacFie and Qiang Wang},
title = {Counting compositions over finite abelian groups},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {2},
doi = {10.37236/7591},
zbl = {1391.05029},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7591/}
}
TY - JOUR
AU - Zhicheng Gao
AU - Andrew MacFie
AU - Qiang Wang
TI - Counting compositions over finite abelian groups
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/7591/
DO - 10.37236/7591
ID - 10_37236_7591
ER -
%0 Journal Article
%A Zhicheng Gao
%A Andrew MacFie
%A Qiang Wang
%T Counting compositions over finite abelian groups
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/7591/
%R 10.37236/7591
%F 10_37236_7591
