THEORY OE THE PARTITION OP NUMBERS. 
661 
the reduced generating function is 
1 
(1 — x) (1 — x 3 ) ’ 
and this is obviously correct, because from the form of the graph we have merely to 
enumerate the ways of partitioning numbers with the parts 1 and 2. 
Art. 55. Again if (/; m ; n) = (2 ; 2 ; go ), we have graphs like 
2 2 
2 2 
2 1 
1 1 
1 
1 
We are led to construct the function 
__ 1 __ 
( j _ «) (i - A) a - *w> (i - £) ’ 
in the expansion of which all terms involving negative powers of a and b have to be 
rejected. Isolating the integral portion and putting a —b — 1, we find the reduced 
generating function 
1 
(1 -X) (1 - x*y (l~- X*) 
a result which, unlike the previous one, is not obvious. 
For the case (l ; m ; n) = (2 ; 3 ; oo ) we introduce additional denominator factors 
(1 _ ctbc3?) (i - £). 
and with increasing labour of algebraical performance we arrive at the reduced 
generating function 
_1_ 
(1 - *) (1 - a»)» (1 — *»)» (1 - «*) ' 
Art. 56- In general for the case 
(l ; m ; n) = (2 ; m ; oo) 
the generating function is the reciprocal of the product of the 2 in factors 
