84 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
Art. 2. Writers upon the problem have not usually observed that the general 
question is identical with the partition of a multipartite number into a given number 
of parts. Thus the problem discussed by Netto and others is simply the 
enumeration of the partitions, into exactly k parts, of the multipartite number 
1111 ... q times repeated. 
Ex. gr. when q = 3, k = 2 (see the two-factor factorization of 2x3x5 above), 
we have three partitions of the multipartite 111 into exactly two parts. These 
partitions are 
( 110 , 001 ), ( 101 , 010 ), ( 011 , 100 ). 
In general the enumeration of the factorizations, involving k factors, of the 
composite number 
' yields the same number as the enumeration of the partitions of the multipartite 
number 
m 1 m 3 ... m t 
into exactly k (multipartite) parts. 
Art. 3. It is the same problem also to enumerate the separations of a given 
(unipartite) partition. Thus in relation to the partition (321) of the number 6, there 
is a one-to-one correspondence between the separations which involve two separates 
and the partitions of the multipartite number 111, which involve exactly two parts. 
The separations are in fact 
(32) (1), (31) (2), (21) (3). 
In general there is a one-to-one correspondence between the separations of the 
partition 
(qrqr • •• 
which involve k separates, and the partitions of the multipartite number 
m x m 2 ... m„ 
which involve exactly k parts. 
Art. 4. The general question of multipartite partition I have already discussed* 
by a method of grouping the partitions and a particular theory of distribution. The 
present investigation which depends upon other principles leads to results of a 
different and more general character. I showed many years agof that in regard to 
the system of infinitely numerous quantities 
®j y, • • ■ > 
* ‘Phil. Trans. Camb. Phil. Soc.,’ vol. xxi., No. xviii., pp. 467-481, 1912, 
t ‘Proc. Lond. Math. Soc.,’ vol. xix., 1887, pp. 220 et seq. 
