624 
MAJOR P. A. MAC MAHON ON THE 
It is also manifest that to each separate of the separation corresponds a part of the 
partition, and that we obtain a partition 
(XiXsXs. •••) 
of the multipartite 
( 7r l 7r 2 7r ,3 * * •)’ 
in correspondence with each separation 
3* 3 ---) (••.)(•••)••• 
of the partition 
{Pi'Pi'Ps* • • •)• 
There is, therefore, identity of enumeration. Also, the enumeration of the separa¬ 
tions into Jc separates is identical with that of the partitions into Jc parts. 
Ex. gr., the subjoined correspondence :— 
Separations. 
Partitions. 
(pV) 
(22). 
(A 3 ) (? 3 )> 
(20 02), 
(p 2 q) (q)> 
(21 ol), 
( p ) 
(to 12), 
( 
(lV). 
(p z ) (#> 
(20 bl 3 ), 
(p? m 
(To 3 02 ), 
(pq) (p) (q)> 
(11 10 01), 
(pf 
(To 3 ol 3 ). 
Art. 7. It is important to take note of the fact that the subject of the separations 
of partitions of unipartite numbers necessitates the consideration of the partitions of 
multipartite numbers. 
The partitions of a multipartite number are divisible into sets in the same manner 
as the separations of a unipartite number. In the case of the number 
( 7r i 77 '2 7r a • • •) 
there are pip. 2 p s ■ • • sets, where p s is the number of partitions of the unipartite tt s . 
