THEORY OF THE PARTITION OF NUMBERS 
623 
In the present instance there are six sets, viz., 
In general, if 
S{(3), (2)}, 
S{(3), (l 3 )}, 
S {(21), (2)}, 
S {(21), (l 3 )}, 
S{(1 3 ), (2)}, 
S{(1 8 ), (]*)}. 
(Px'Pz'Ps* • • •) 
be the separable partition, the multiplicity is 
( n i 77 2 77 ?, • • •) I 
and if the unipartite v s possess p s partitions, the separations can be arranged in 
Pi Pi P 3 • • • 
sets. 
I observe that the notion of sets of separations enters in a fundamental manner 
into the theory of symmetric functions. 
Art. 6. One of the first problems encountered in the arithmetical theory is the 
enumeration of the separations of a given partition. I shall prove that the number 
of separations of the partition 
(P-PPi'P*' • • •) 
is identical with the number of partitions of the multipartite number 
( 77 T 7J V 7r 3 • • •)> 
formed from the multiplicity of the separable partition.* 
If we separate (p\ l p£ 2 Pz* .. .) so that one separate of the separation is 
• • •)> 
it is clear that we can partition the multipartite (Tr^rr^TT^. . .} in such wise that one 
part of the partition is 
fXiX-iXs • • •)• 
* Note that the x’ule 
partite number. 
distinguishes a multipartite number from a partition of a uni- 
