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

\[gf(2k) = p(0) + p(1) + p(2) + \cdots + p(k-1)\]

where \(g_f(2k)\) 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 \(j\).