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



Art. 7. Let N(&c...) denote the number of permutations of the first n integers 

 which have a descending specification denoted by the composition 



(ale...) 

 of the number n. 



Obviously N ()=!, a = n. 



To determine N (ab), a + b = n, separate the n integers into two groups, a left-hand 

 group of n numbers chosen at random and a right-hand group of the remaining b 

 numbers. This can be done in 



( ) different ways. 

 W 



I write . r-7 = 1 j in a common notation ; now arrange each group of 



numbers in descending order of magnitude for each of the ( ) separations ; we thus 



obtain each of the permutations enumerated by N (a, b) and the one permutation 

 enumerated by N (a + b). 



Hence 



or 



~ \aj \a + bj \a 



Again, to find N (abc), we separate the n integers into three groups containing 

 a, b, and c integers respectively ; this can be done in 



a\b\c\ 



different ways ; placing the numbers in each group in descending order., we obtain all 

 the permutations enumerated by 



N (abc), N (a + b, c), N (a, b + c), N (a + b + c). 

 Hence 



N (6c) + N (a + b, r) + N (a, 6 + c) + N (a + b + c) = ^[^ ; 

 leading to 



nl nl 



T 



a! 6! c\ (a + b)l c! a! 

 where a+b + c = n. 

 Similarly we find 



n! n\ n\ n\ 



N (abod) = 



a\b\c\d\ 



(ct+b)l(c+d)l 

 where a + b + c + d = n, 



n ! n! n ! n ! 



