658 
MAJOR P. A. MACMAHON OR THE 
a given number of nodes, corresponding to the regularised partitions of all multi¬ 
partite numbers of given content, is a weighty problem. I have verified to a high 
order that the generating function of the complete system is 
(1 — x) 1 (1 — x 3 ) 3 (1 — x 3 ) 3 (1 — a? 4 ) 4 . . . ad inf., 
and, so far as my investigations have proceeded, everything tends to confirm the 
truth of this conjecture. 
I observe that, to negative signs pres, the exponents are 
1, 2, 3, 4, 5, . . . 
viz., the figurate numbers of order 2. 
The generating function which enumerates the two-dimensional graphs, is 
(l — o’) -1 (1 — x 3 ) -1 (1 — x 3 ) -1 (l — x 4 ) -1 . . . 
where (notice) the exponents are 
1 , 1 , 1 , 1 , 1 , . . . 
the figurate numbers of order 1. 
Proceeding further back, we find that one-dimensional graphs are enumerated by 
(1 — x)~ l (1 — x 2 ) 0 (1 — a; 3 ) 0 (1 — x 4 ) 0 . . . 
the numbers 
1 , 0 , 0 , 0 , 0 , . . . 
being the figurate numbers of order zero. Going forward again it is easy to verify up 
to a certain point that four-dimensional graphs (which it is quite easy to graphically 
realise in two dimensions) are enumerated by 
(1 — a 1 ) -1 (1 — x 3 ) -3 (1 — X’ 3 ) -0 . . ., 
where the exponents involve the figurate numbers of order 3. 
The law of enumeration appears, conjecturally, to involve the successive series of 
figurate numbers. 
Art. 52. Before proceeding to establish certain results, it may be proper, as illus¬ 
trating the method pursued in this difficult investigation, to give other results which, 
at first mere conjectures, are gradually having the mark of truth stamped upon them. 
Consider graphs in which only the numbers 1 and 2 appear. These are two-layer 
partitions. The enumeration to a high order is given by the generating function 
(2 
oo ; oo ) — (1 — .r)- 1 (1 - x 2 )- 2 (1 - X 3 )- 3 (1 - X 4 )- 3 . . . 
