On the Number of Graphical Forest Partitions

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

Research output: Contribution to journalArticle

1 Citation (Scopus)

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

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Partition
Integer Partitions
Degree Sum
Graphics
Simple Graph
Partition Function
Count
Disjoint
Union
Integer
Subset
Graph in graph theory
Vertex of a graph

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Frank, D. A., Savage, C. D., & Sellers, J. A. (2002). On the Number of Graphical Forest Partitions. Ars Combinatoria, 65, 33-37.
Frank, Deborah A. ; Savage, Carla D. ; Sellers, James Allen. / On the Number of Graphical Forest Partitions. In: Ars Combinatoria. 2002 ; Vol. 65. pp. 33-37.
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Frank, DA, Savage, CD & Sellers, JA 2002, 'On the Number of Graphical Forest Partitions', Ars Combinatoria, vol. 65, pp. 33-37.

On the Number of Graphical Forest Partitions. / Frank, Deborah A.; Savage, Carla D.; Sellers, James Allen.

In: Ars Combinatoria, Vol. 65, 01.10.2002, p. 33-37.

Research output: Contribution to journalArticle

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N2 - 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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Frank DA, Savage CD, Sellers JA. On the Number of Graphical Forest Partitions. Ars Combinatoria. 2002 Oct 1;65:33-37.