THEORY OF THE PARTITION OF NUMBERS. 
G 57 
fi ( x )>A( x )’fs (x) being the generating functions for the essentially distinct graphs 
which are symmetrical, qnasi-symmetrical, and unsymmetrical respectively. 
x41so we may write 
F (x) = F 1 (x) + F 3 (x) + F s (x), 
where F, (x), F 3 (x), F 3 (x) are the generating functions of the partitions of the three 
natures. 
The present theory is really the solidification of Sylvester’s theory given in the 
‘American Journal of Mathematics’ ( loc . cit.). Already we have seen that the 
Sylvester-graphs are susceptible of a far wider interpretation than was at first antici¬ 
pated. If we view these graphs from a two-dimensional standpoint, every graph is 
either symmetrical or unsymmetrical, the symmetrical class comprising all graphs 
which are self-conjugate. If, however, our standpoint be three-dimensional, there are 
no longer any symmetrical graphs. The two classes are the quasi-symmetrical and the 
unsymmetrical. A single exception to the above occurs where the graph is of unity. 
Moreover, the classes now do not comprise the same members. Certain graphs which 
were unsymmetrical from the first standpoint appear as quasi-symmetrical from the 
second. 
Omitting the trivial symmetrical graph of unity every two-dimensional graph 
can be read either in three or six ways. The quasi-symmetrical class giving three 
readings, comprises the self-conjugate graphs and also those which consist of either 
a single line or a single column of nodes. The remaining graphs give six readings. 
Ex. gr. The graph 
11111 
yields the three partitions (5), (11111), (11111) being quasi-symmetrical from the 
three-dimensional standpoint although it is unsymmetrical in Sylvester’s theory. 
Also the self-conjugate graph 
Ill 
1 
1 
yields the three partitions (311), (311), (111 100 100). 
Such a graph as 
111 
111 
being unsymmetrical in both theories yields six partitions 
(33), (33), (222), (222), (mill), (Tl TT IT). 
Art. 51. The enumeration of the three-dimensional graphs that can be formed with 
MPCCCXCVL — A, 4 P 
