For a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.
@article{10_37236_5819,
author = {Olivier Bernardi and Caroline J. Klivans},
title = {Directed rooted forests in higher dimension},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {4},
doi = {10.37236/5819},
zbl = {1353.05131},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5819/}
}
TY - JOUR
AU - Olivier Bernardi
AU - Caroline J. Klivans
TI - Directed rooted forests in higher dimension
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/5819/
DO - 10.37236/5819
ID - 10_37236_5819
ER -
%0 Journal Article
%A Olivier Bernardi
%A Caroline J. Klivans
%T Directed rooted forests in higher dimension
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/5819/
%R 10.37236/5819
%F 10_37236_5819
Olivier Bernardi; Caroline J. Klivans. Directed rooted forests in higher dimension. The electronic journal of combinatorics, Tome 23 (2016) no. 4. doi: 10.37236/5819
