66 MAJOR P. A. MA. MAHON ON THE COMPOSITIONS OF NUMBKKS. 



The first of the two questions has given rise to two new symmetric {'unctions, 



of great interest, which supply the complete solution. The second gives rise to the 

 same generating function .that presented itself in the first Memoir. It is here 

 attacked by the calculus of symmetric function differential operators, and a number 

 of new results obtained. 



If the whole pack be specified by the partition 



there is a one-to-one correspondence between the arrangements which lead to a 

 distribution into m packs and the principal compositions, involving m 1 essential 

 nodes, of the multipartite number 



~~ 



Part I. is concerned with an elementary theory of the case in which the cards are 

 all numbered differently. 



The general case, which is more difficult, is dealt with in Part II. 



To make what follows clear to the reader, I commence with some elementary 

 notions concerning the connection between the partitions and compositions of 

 numbers on the one hand, and permutations and combinations of things on the other 

 hand, and I also specify and describe the nomenclature and notation that I have 

 found it convenient to adopt. A suitable notation is, indeed, of the first importance 

 in this subject, as I hope to make evident as the investigation proceeds. 



INTRODUCTORY. 



Art. 1. Any succession of numbers, written down from left to right at random, 



such as 



142771, 



is termed a " composition " of the number which is the sum of the numbers. 

 If the numbers be arranged in descending order from left to right, 



774211, 



the succession is termed a "descending partition," or simply a "partition" of the 

 number which is the sum of the numbers. 



Or, if we arrange in ascending order of magnitude, 



112477, 



the succession may be termed an " ascending partition." 



Generally, in speaking of partitions, we understand that the descending order is 

 meant ; but it is convenient sometimes to consider them as being defined by an 

 ascending order. 



