On the Number of Graphical Forest Partitions


A graphical partition of the even integer n is a partition of n where each part of the partition is the degree of a vertex in a simple graph and the degree sum of the graph is n. In this note, we consider the problem of enumerating a subset of these partitions, known as graphical forest partitions, graphical partitions whose parts are the degrees of the vertices of forests (disjoint unions of trees). We shall prove that gf(2k) = p(0) + p(1) + p(2) + . . . + p(k − 1) where gf(2k) is the number of graphical forest partitions of 2k and p(j) is the ordinary partition function which counts the number of integer partitions of j.

Extracted Key Phrases

Cite this paper

@article{Frank2002OnTN, title={On the Number of Graphical Forest Partitions}, author={Deborah A. Frank and Carla D. Savage and James A. Sellers}, journal={Ars Comb.}, year={2002}, volume={65} }