MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS.. 79 



which, by the multiplication theorem, may be given the form 



+ N ( ty, . ./>., 1 '</, . . . 



+ N ( \p, . . .p m> 2 qi . . .c/^r, . . .r m \ )]. 



Evidently, from the above, comprehensive results can be obtained from the 

 multiplication theorem. 



SECTION 2. 



Art. 22. The next problem I propose to solve is that of determining the number of 

 the permutations of the first n integers, whose descending specifications contain a 

 given number of integers, or, in other words, whose associated compositions involve a 

 given number of parts. The solution is implicitly contained in a paper I wrote in the 

 year 1888.* 



Let N., denote the number of permutations associated with compositions containing 

 exactly m parts. 



In the paper quoted, I had under view a collection of objects of any species say 

 p of one sort, q of a second sort, r of a third, and so on and defined the objects as 

 to species by these numbers placed in brackets. I thus formed a partition 



(pqr...) 



of the number n, such partition being the species definition of the objects. 



As equalities may occur between the numbers p, q, r, ..., I took, as a more general 

 definition, the partition 



(pfW~), 



where zirp = n. 



In the case under consideration, where the integers (or objects) are all different, 



the species definition is the partition 



(I'). 



I proved, in the general case, that the number of ways of distributing the objects, 

 into m different parcels, is given by the series 



" /m+p t -l\* /m+p 3 -l\" 

 \ P, ) \ P* I ' 



(m\ /m+p 1 -2\' 1 /m+p t -2\*' fm+p t -2\' 



"\ ft 



(m\ /m+^,-3\- /m+^,-3\" /m+p,-3\" 

 Wl Pi ) \ P, ) \ P* I 



" Symmetric Functions and the Theory of Distributions," Proc. L. M. S.,' Tol. *., p. 225. 



