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XVII. Memoir on the Theory of the Compositions of Numbers. 
By P. A. MacMahon, Major R.A., F.R.S. 
Received November 17—Read November 24, 1892. 
§ 1. Unipartite Numbers. 
1. Compositions are merely partitions in which the order of occurrence of the parts 
is essential; thus, while the partitions of the number 3 are (3), (21), (111), the com- ' 
positions are (3), (21), (12), (111). 
The enumerations of the compositions of a number n into p parts, zeros excluded, 
is given by the coefficient of x” in the expansion of 
(a: -f -f- . . .)^; 
this expression may be written 
and the coefficient of x”‘ is seen to be 
^ _ 1\ * 
p - ij • 
The generating function of the total number of compositions of n is 
£ (a; + -{-...)/’ = p -- , 
1 X. *“ 
hence the number in question is 
2”-h 
2. If the parts of the compositions are limited not to exceed s in magnitude, the 
generating function of the number into p parts is 
(x + a:2 + + . . , + a^)p = a;^ 
( W’\ ^ 
) for ———^. 
pl p ! (71 - p) ! 
5 0 2 13.11,93 
