Graphical Basis Partitions

A partition of an integer n is graphical if it is the degree sequence of a simple undirected graph It is an open question whether the fraction of partitions of n which are graphical approaches as n approaches in nity A partition is basic if the number of dots in its Ferrers graph is minimum among all partitions with the same rank vector as In this paper we investigate graphical partitions via basis partitions We show how to e ciently count and generate graphical basis partitions and how to use them to count graphical partitions We give empirical evidence which leads us to conjecture that as n approaches in nity the fraction of basis partitions of n which are graphical approaches the same limit as the fraction of all partitions of n which are graphical