On the Number of Graphical Forest Partitions

Deborah A. Frank, Carla D. Savage, James Allen Sellers

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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.

Original languageEnglish (US)
Pages (from-to)33-37
Number of pages5
JournalArs Combinatoria
Volume65
StatePublished - Oct 1 2002

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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    Frank, D. A., Savage, C. D., & Sellers, J. A. (2002). On the Number of Graphical Forest Partitions. Ars Combinatoria, 65, 33-37.