844 
MAJOR P. A, MACMAHON ON THE THEORY 
compositions and the number comprised of r parts of the multipartite number 
P 1 P 2 P 3 • • • ; so that, 
F {PiVilh . ■ .) = . . ., 
Writine’ 
(1 — a^) (1 — ag) (I — ttg) . . . = 1 ■— + ^2 — «3 + . . . 
+ 7^2 + . . . _ 4” ■ 
1 — 7^0 7^2 . . • 1 2 — cCtt ”f~ Cg — , . b 
This new form of the generating function gives a relation connecting the number of 
compositions of any multipartite with numbers related to lower multipartites. 
For unipartite numbers 
rdw. = sFfe)v'. 
yielding 
F(l)=l 
and 
F = 2F — 1) when > 1. 
For bipartite numbers 
«1 + — ajKj 
1 — 2 («j + a, — a^aj) 
^ F (yiPo) a/>a/^ 
giving 
F {pdh) = 2F (pj - 1, P 2 ) + 2F (pi, Pa - 1) - 2F (p^ - 1, p^ - 1), 
and similarly for multipartite numbers 
F {PilMh . . .) = 2 {F (pi - 1, Pa, Pg, 
— 2 {F (pj — 1, Pa — 1, Ps, ‘ J 
+ 2 {F (pi — 1, Pa — 1 Pg — 1, . . 
a formula absolutely true for all multipartites superior to ]11 . , and universally 
true if F (000 . . .) when it occurs be interpreted to mean 
12, A simple expression is obtainable for F (piPs). We have to find the coefiicient 
of ill 
«! + ag — ai«2 
1 — 2 (aj -f 1 X 2 ~ «l“2) 
or, this is the coefficient of in 
2P2-1 ~ — 
(1 - 2«ip^+i ’ 
* See post, Art. 39. 
