640 
MAJOR P. A. M ACM AH ON ON THE 
For the enumeration we must take the coefficient of a s x sp ( 2 ) in the development 
of the generating function 
axP 
1 _ (1 + ax) (1 -f- cix 2 ) (1 + ax 3 ).... (1 + ax p x ) ; 
1 . J. 
or the coefficient of (ax? is ) s 1 in 
1 
1 — X 
(1 + ax) (1 + ax 2 ) (1 -f- «x 3 ) . . . (1 + axP ')• 
But the number of compositions of p into 5 parts is 
'p - V 
s — 1 / ’ 
and thence we see that a term in the development of 
-—- (1 + ax) (1 + ax 2 ) (1 + ox 3 ) . . . (1 + ax p ’), 
is 
p — 1 
s — 1 
qS-1 2q(p-Is)(s- 1 ) . 
and, giving s successive values, we can isolate, from this product, a portion 
(P ~ ™2~,2n-3 1 (P ~ A „.3~.3jj- 6 i i „p-l ™(f) 
1 + b’ 1 " ) ax v 1 + 0 " ) a 2 x- p ~ z -j- [ l> ,, a 3 x 3jl 6 -f . . . + a p 1 x w ; 
P ~ 1 
1 
or symbolically (1 -f- ax p ) p ~ l , where after expansion x w> is to be replaced by x pp ~- p(p+1) . 
Art. 28. By leaving s unspecified we can readily reach a theorem concerning the 
product, 
(1 + x) (1 -{- x 2 ) (]. + x 3 ) . . . (1 -f x p ~ l ). 
(p\ 
It is easy to show that the coefficient of x yl ', in the development, is 2 i>_1 . We may 
say that the number of partitions of (^ ) and all lower numbers into unrepeated parts 
not exceeding p — 1 in magnitude is For p = 5 these partitions are : 
4321 
432 
43 
31 
4 
431 
42 
21 
3 
421 
41 
2 
321 
32 
1 
, 16 in number. 
Art. 29. It is obvious that, by the same process, we can obtain a correspondence 
