The Enumeration of Graphical Partitions

Abstract

The present paper greatly extends and elaborates the results of a previous paper by Stein on the enumeration of graphical partitions. The latter are a subset of partitions π ⊢ 2 q with exactly p non-zero parts, with q and p respectively the number of lines and points of a graph without multiple lines, loops or isolated points. The enumeration is effected in terms of an auxiliary set of partitions λ which are in 1-1 correspondence with an appropriate subclass of the partitions π. It is shown, inter alia, how Hakimi's algorithm imposes a definite recursive structure on the set {λ}. It is also shown how to write down in principle explicit enumeration formulae for any r ≡ q − p .