P. A. M^MAHON ON THK coMpasiTioNs OF NUMBERS. 89 



For it was shown, lor. cit., that the number of ways of distributing the objects, 

 as specified, into different parcels containing a, b, c... objects respectively is the 



coefficient of the symmetric function 



(abe...) 



in the development of the symmetric function 



as a sum of monomial symmetric functions. 

 Let this coefficient lie denoted by 



and let the number of arrangements of the objects, which have a descending 

 specification 



be denoted by 



Let the whole number of objects be 



tirp = n. 

 Then, when a n, clearly 



N(a) = C(a)= 1, 



and when a + b = n, C(ab) is the number of arrangements into two different parcels 

 containing a, b objects respectively, and by previous reasoning 



and, when a + b+c = n, 



N (dbc) + N (a + 6, o) + N (a, b + c) + N (a + b + c) = G (abc), 



and so forth as in the simple case already considered. 

 Hence 



N (oft) C(ofr)-O (<>+&), 



N (abc) = C (abc)-C (a+b, o)-C (a, b+c)+C (a+b+c), 

 N (abed) = C (abcd)-G (a + b, c, d)-C (a, b + c,d)-C (a, l>, c+d) 



Ac., 



the numbers N being all expressible iu terms of coefficients of the auxiliary generating 

 function. 



Art. 34. E.g. Take objects aaa/8/Jy, where a, ft, y are in descending order of 

 magnitude. 



Since 



hMi = (6) + 3 (5l) + 5 (42)4-8 (41 3 ) + 6 (3*)+ 12 (321) 



+ 1 9 (3 1 8 ) + 1 5 (2) + 24 (2 1 !*) + 38 (2 1*) + 60 ( 1), 



VOL. CCVJI. - A. N 



