670 
MAJOR P. A. MACMAHOH OR THE 
function for the number of two-layer graphs in which the number of rows is limited 
to unity; that is, of the graphs specified by (2 ; 1 ; oo ). The graph either does 
not or does contain a specification graph line. The former are enumerated by 
the latter by x 2 k x ( x ). 
Hence, 
or 
3 
/c, ( x ) = - ^ 4- x 9j k l ( x) 
ki W = (1 _ X ) (l _ P) ; 
and the number of specification graphs containing one graph line is 
x°k x ( x ) or 
(1 - x) (1 - X*) 
Next let Jc. 2 (x) denote the number of two-layer graphs in which the number of rows 
is restricted to two. If it contain no specification graph line it must be of the form 
1 K 
P X ; lx. 
For these the generating function is 
1 
(1 — x ) (1 — x 2 ) 
If it contain one specification graph line it must be of the form 
2 K +1 ]_/“■ 
n „ (X + /X + 1 > v) 
and these are enumerated by 
x~k x ( x) 
1 — x 
If it contain two specification graph lines its form must be 
2* + 2 
2*' + i j A 
