108 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 



Consider the expression ; 7 



n (pg...) 'hf9...Y> 



which is the generating function for the difference between 



N(pq...) and ~N (pq...)', 



for all specifications of the numbers permuted. 

 The generating function may be written 



^P7...~W..> 



according to the theorem proved above. 

 The differential operation ^ 



has the equivalent forms 



hence n , 



A "- 



is the same function of , , , 



flj, fl a , h a , ... 



that 



D!^... 



is of 



1, 2, 3, 



It follows at once that ^. ,, \ 



Di"(*--W-) - 



equivalent to the known result 



already found. 



Art. 66. Now, considering the generating functions 



or 



must be of the form , T>L a 



(A + 2B){(2) + 2(1 2 )}. 

 Hence r -sr 



... 

 equivalent to 



N (p 9 ...) (21 ... +N (pq.-.Y^^ = N Qjgr.. .)(,. 



Thus, from the table n = 6, 



(3 

 13 -f 6 = 19. 



