THEORY OF THE PARTITION OF NUMBERS. 
625 
The same succession of numbers may be employed to denote either a multipartite 
number or the partition of a unipartite number, so that we naturally find great 
similarity between the theories of unipartite partitions and of multipartite numbers. 
We see above that the separations of partitions of unipartites are in co-relation with 
the partitions of multipartites. 
Art. 8. In a natural manner the separations of a partition of a multipartite present 
themselves for consideration. In correspondence we find what may be termed a 
double separation of a partition of a unipartite. 
Ex. gr. Consider the partition (20 01 3 ) of the multipartite (22), which is co¬ 
related to the separation ( p z ) (q ) 3 of the partition ( p^q 1 ) of the unipartite 2 p + 2 q. 
Of this partition we find a separation 
(20 01 ) ( 01 ) 
in correspondence with a double separation 
l(p 2 ) ( r /)> (?)} 
of the partition (p 2 g 2 ). 
Hence the enumeration of the separations of a multipartite partition is identical 
with that of the double separations of a unipartite partition. 
In general n-tuple separations of unipartite numbers correspond one-to-one with 
n — 1-tuple separations of multipartite numbers. 
For the present I leave the subject of separations, merely remarking that the 
theory was made the basis of all the memoirs on symmetric functions to which 
reference has been given, and that algebraically considered they are of extraordinary 
interest. 
§ 2. The Graphical Representation of Partitions. 
Art. 9. In an important contribution to the theory of unipartite partitions 
Sylvester* adopted a graphical method which threw great light on the subject, and 
was fruitful in algebraical results. 
The method consisted in arranging rows of nodes, each row corresponding to a part 
of the partition and containing as many nodes as the number expressing the magni¬ 
tude of the part. 
Ex. gr., the partition (32 2 1) has the graph 
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* “A Constructive Theory of Partitions," by J. J. Sylvester, with insertions by Ur. F. Franklin, 
‘ American Journal of Mathematics,’ vol. 5. 
MDCCCXCVI.—A. 4 L 
