We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus $2$ partitions, after identifying all genus $2$ primitive partitions, using a computer-assisted search.
@article{10_37236_7632,
author = {Robert Cori and G\'abor Hetyei},
title = {Counting partitions of a fixed genus},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {4},
doi = {10.37236/7632},
zbl = {1402.05101},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7632/}
}
TY - JOUR
AU - Robert Cori
AU - Gábor Hetyei
TI - Counting partitions of a fixed genus
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/7632/
DO - 10.37236/7632
ID - 10_37236_7632
ER -
%0 Journal Article
%A Robert Cori
%A Gábor Hetyei
%T Counting partitions of a fixed genus
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/7632/
%R 10.37236/7632
%F 10_37236_7632
Robert Cori; Gábor Hetyei. Counting partitions of a fixed genus. The electronic journal of combinatorics, Tome 25 (2018) no. 4. doi: 10.37236/7632
