On the Number of Graphical Forest Partitions

Abstract

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.

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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} }