520 
MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
the summation being for every separation 
{pn^ir“ Pn(in'^ ■ • .. .y'u. - 
of the partition • ■)• (Compare Art. 43 .) 
Observe that in these formulm the multiplications of operations are non-symbolic 
and denote successive operations. 
67 . Remark the results of operations :— 
j\ h 
9 (Pilin’^” ■••) 9 • • •) • • • 
• • • {lhiqn^^lhz9n'^- • • • • . • = 1 . 
J. 
iiOs! 
§11. The Multiplication of Symmetric Functions. 
68. The partition G operators are of great service in multiplication. An example 
will make this clear. It is required to find tlie coefiScient of (11^) in the product 
(T02) (01)2. 
Put 
(Ib2) (01)2 = .,. + A (Ti2) + ... 
2 
On operating with on the right the result is A, since every other term is 
annihilated ; and since 
C]^;^ = C(n) “h G(JoOT); 
we have 
Gn (Tb2) (bd)2 = {G(n) + Girom)VC^O^) (^)' = A, 
therefore 
{C(n) + G(iooi)}. 2 (id) (01) = A, 
hence 
A = 2. 
Similarly putting 
(PifiiiP" • • •) {PMhf'PnFl" ...)(...) = ... + A (^qs/^ ...)+••• 
we have merely to operate on the left-hand side with the partition operators 
Pi P2 ^ 
equivalent to G^^^^ G^^^^ ... in order to find A. 
