1892.] Theory of the Compositions of Numbers. 291 



Section treats of the compositions of unipartite numbers both analyti- 

 cally and graphically. The subject is of great simplicity, and is 

 only given as a suitable introduction to the more difficult theory, 

 connected with multipartite numbers, which is developed in the 

 succeeding sections. 



The investigation arose in an interesting manner. In the theory 

 of the partitions of integers, certain partitions came under view 

 which may be defined as possessing the property of involving a 

 partition of every lower integer in a unique manner. These have 

 been termed "perfect partitions," and it was curious that their 

 enumeration proved to be identical with that of certain expressions 

 which were obviously " compositions " of multipartite numbers. 



The 2nd Section gives a purely analytical theory of multipartite 

 numbers. 



is the notation employed in the case of the general multipartite 

 number of order n. The parts of the partitions and compositions of 

 such a number are themselves multipartite numbers of the same 

 order. Of the number 21 there exist 



Partitions. Compositions. 



(20 01) (20 01), (01 20) 



(II 61) (II 16), (16 ID 



(I6 2 61) (IS SDi (16 61 16) i (61 16*)- 



The generating function which enumerates the composition has the 

 equivalent forms 



l2(a i aza 3 .. .) 



where h st a s represent respectively the sum of the homogeneous 

 products of order * and the sum of the products s together of 

 quantities 



*1> 2> S ) * 



and the number of compositions of the multipartite ' 



is the coefficient of at^o^f*.. .**" in the development according to 

 ascending powers. 



