66G 
MAJOR P. A. MACMAHON OX THE 
Ex. cjr. take (l ; to ; n) — {2 ; 2 ; l); the fraction is 
__1_ 
(i - m x) (i - ■£) a - (i - 
We have to retain the integral portion of 
(i+^+iw 2 )(i + ” + ~) ; 
selecting this and putting^ = q { = 1, we obtain 
1 -f- cr 4- 2x 2 + ;r 3 + ad, 
which is 
(1 - x~) (1 - ar 5 ) 2 (1 - ad) 
(1 _ x) (1 _ * 2)3 (1 _ * 3 ) • 
As in simpler cases I have not been able to overcome the algebraic difficulties, it 
is perhaps needless to say that in this most general case I cannot establish the form 
of the reduced generating function. 
Art. 60. I return to consider various particular points of the problem. When the 
number of' layers of nodes is restricted to two, we have seen that the generating 
function which enumerates the graphs that can be formed with a given number of 
nodes is 
(1 — x)~ l (1 — x 3 ) -2 (l — cc 3 ) -3 (1 — ad) “ 3 . 
In correspondence we have the regularised bipartitions (including uni-partitions) of 
multipartite numbers of given content. 
Also if the breadth of the graph do not exceed to or the multipartite numbers be 
not more than TO-partite the generating function is 
(1 — x)~ l {(1 — x~) (1 — x 3 ) . . . (1 — x m )}~ 2 (l — cc" l+1 ) -1 . 
I propose to give another proof of these results based upon a certain mode of dis¬ 
section of the graph. 
In the notation that has been used, a graph may be written 
2 a p 
2 V F' 
2 a" p" 
