### 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 language | English (US) |
---|---|

Pages (from-to) | 33-37 |

Number of pages | 5 |

Journal | Ars Combinatoria |

Volume | 65 |

State | Published - Oct 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Ars Combinatoria*,

*65*, 33-37.

}

*Ars Combinatoria*, vol. 65, pp. 33-37.

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the Number of Graphical Forest Partitions

AU - Frank, Deborah A.

AU - Savage, Carla D.

AU - Sellers, James Allen

PY - 2002/10/1

Y1 - 2002/10/1

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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0142169033&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0142169033&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0142169033

VL - 65

SP - 33

EP - 37

JO - Ars Combinatoria

JF - Ars Combinatoria

SN - 0381-7032

ER -