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



on the right-hand side, is not equal to any other number in the same bracket ; so 

 that, when *<%n, (g+ J} ^ {(n _ g) (} }>} 



Hence 



J m = ( 

 K m = (n-m+1) N {(m) (l) }-2 (n-m) 



and, the specification of the numbers permuted being 



(pqr...), 



N {(a) (b) (c) . . . } = C (abc. . .) = DJ) 6 D C . 

 thence T 



or, m not being less than the greatest integer in (it + 1 ), 



is the generating function of the number J,,, . 

 Similarly ^ 



and, m not being less than the greatest integer in \ (+ 1), 

 (n-m+ 1) /< m /t 1 "- m -2 (n-wi) ft*+A""*~ 1 t(-W 



is the generating function of the number K m . 



Similarly, but subject now to the condition that m must not be less than the 

 greatest integer in (n 1), 



(_,_ |) /-, n+ A"~"" 2 -(-m) A.+A"'"' 1 

 is the function which generates the number 



Subject to the conditions mentioned, we have a complete solution of the problem, 

 but when m has other values, the solution is less simple and I see no way of 

 effecting it. 



