80 MAJOR P. A. MAcMAIION ON THE COMPOSITIONS OF NUMBERS. 



For the case in hand, p^ = 1,77] = n, 



Art. 23. I shall prove that 



For consider the arrangements enumerated by F m . Place the compartments (or 

 parcels) in order, from left to right, in any one such arrangement, and, in each 

 compartment, place the integers in descending order of magnitude. The arrangement 

 is obviously one of those enumerated by 



N m) N m _j, N m _ 2 , ... or N,. 



In the whole of the arrangements, enumerated by F m , thus treated, each arrange- 

 ment enumerated by N m will occur once only. 



1 | 2 | 3 | 4 | \rn-l or ms. 



. ._. . .......... -_. 



Let the illustration denote an arrangement enumerated by N m _!. Each segment 

 denotes an integer, and the m 2 vertical lines separate the integers into com- 

 partments. 



By placing an extra vertical line at one of the unoccupied points of division, we 

 obtain an arrangement enumerated by F m . This can be done in (n 1) (m 2) 

 different ways, showing that the particular arrangement, enumerated by N ra _j, is 

 derivable by obliteration of a vertical line from 



n m+l 



different arrangements enumerated by F m . 



Hence, the forms F m include the forms N OT _j each n m+l times. 



Again, let the illustration denote an arrangement enumerated by N m _,. By placing 

 s extra vertical lines, at unoccupied points of division, we obtain an arrangement 

 enumerated by F m . This can be done in 



i'nm+s\ 



\ * / 



different ways; showing that the particular arrangement, enumerated by N m _,, is 

 derivable, by obliteration of ,s- vertical lines, from 



nm+s\ 

 s 



different arrangements enumerated by F m . 



