THEORY OF THE PARTITION OF NUMBERS. 
669 
Ex. gr. Suppose the specification graph to be 
2 2 2 2 
2 2 1 
2 
and the unipartite partition to be 
(9 7 7 6 5 5 4 3 2 2). 
Interpreted on the line of route concerned, which is that marked upon the scheme 
above, we obtain 
2 2 2 2 1 
S 2 2 2 2 
2 2 2 1 
2 2 2 1 
2 2 11 
2 2 1 
2 2 1 
S 2 2 1 
2 1 1 
2 1 
2 
2 
S 2, 
in which the specification graph lines, marked S, have been interpolated. 
Hence, if F (x) be the generating function which enumerates specification graphs, 
__FM_ 
(1 — x ) (1 — x 3 ) (1 — x 3 ) (1 — x*) .. . ad inf. 
will be the generating function of all two-layer graphs—that is of forms specified by 
(2 ; co ; oo ). 
We have next to determine the form of F ( x). 
Art. 62. A specification graph may contain no graph lines ; this will be the case when 
the line of route through the scheme is the lowest horizontal line. There is only one 
such graph ; generating function 1. If it contains one line, this line must be of the 
form 2 A P (X > 0), and the number of such graphs is given by the generating function 
x? 
(1 — x) (1 — X 2 ) 
We may also take the following view of the matter. Let k 1 (») be the generating 
