90 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS, 



we calculate, from the above formulae, 



N(6)=l, ' . ' - 



N (51) = 3-1 = 2, 

 N(3 2 ) = 6-l = 5, 



N(321) = 12-5-2-1 = 4 

 and so on. 



The five arrangements, enumerated by N (3 a ), are 



OLa. 



.ya.pfi 



a/3/3aay 

 /3/3yaaa, 



each having the descending specification (3 a ). 



The four arrangements, enumerated by N (321), are 



each having the descending specification (321). 



The complete results qud numbers specified by (321) are 



N (31 3 ), N (1 3 3), N (21 4 ), N (1 4 2), N (2112) } 



>each . . 

 N (121 3 ), N (1 3 21), N (1 2 21 2 ), N (I 6 ) J 



N (6), N (41 2 ), N (1 2 4), N (131 2 ), N (1 2 31)| 

 N (2 2 1 2 ), N (1 2 2 2 ), N (2121), N (1212) J 



N(51), N(15), N(312), N(213) 2 . . 8 



N (141), N (1221) 3 . . 6 



N(42), N(24), N(321), N(123) 4 . , 16 



N(3 2 ), N (231), N (132) 5 . . 15 



N(2 :1 ) 6 . . 6 



60 



60 being, of course, the total number of permutations of the objects. 



