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XVI. Memoir on the Theory of the Partition of Numbers. — Part I. 
By Major P. A. MacMahon, R. A ., F.R.S. 
Received December 31, 1895,—Read January 16, 1896. 
§i- 
Art. 1. I have under consideration multipartite numbers as defined in a former 
paper.'" 
I recall that the multipartite number 
a/3y 
may be regarded as specifying a -j- /3 -f- y + . . . things, a of one sort, /3 of a second, 
y of a third, and so forth. If the things be of m different sorts the number is said to 
be multipartite of order m or briefly an m-partite number. It is convenient to call 
a, (3, y,.. . the first, second, third. . , figures of the multipartite number. If such a 
number be divided into parts each part is regarded as being m-partite; if the order 
in which the parts are written from left to right is essential we obtain a composition 
of the multipartite number; whereas if the parts themselves are alone specified, and 
not the order of arrangement, we have a partition of the multipartite number. This, 
and much more, is explained in the paper quoted, which is concerned only with the 
compositions of multipartite numbers. 
Art. 2. The far more difficult subject of partitions is taken up in the present paper. 
The compositions admitted of easy treatment by a graphical process. An ■m-partite 
reticulation or lattice is taken to be the graph of an m-partite number, and on this 
graph every composition can be satisfactorily depicted. 
A suitable graphical representation of the partitions appears to be difficult of 
attainment. As the Memoir proceeds, the extent to which the difficulties have been 
overcome will appear. There are several bonds of connection between partitions 
and compositions; in general, these do not exist between m-partite partitions and 
m-partite compositions, but arise from a general survey of the partitions and compo¬ 
sitions of multipartite numbers of all orders. 
These bonds are of considerable service in the gradual evolution of a theory of 
partitions. Two bonds have already been made known ( loc . tit.). They both have 
* “Memoir on the Theory of the Composition of Numbers,” ‘Phil. Trans.,’ R.S. of London, vol. 184 
(1893), A, pp. 835-901. 
4 K 2 
9.1.97 
