656 
MAJOR P. A. MACMAHON ON THE 
Number 4. 
«•«» ©»@ (•) ® 
@ ® • • 
and the whole of the partitions are obtainable by reading them in the various ways 
above explained. 
Art. 49. It will be convenient to adopt in future another notation for the graphs; 
the number m will denote a vertical column of m nodes piled upon one another. 
The 13 graphs appertaining to the number 4 are written 
1111 
111 
2 L1 
11 
21 
11 
3L 
1 
1 
1 
11 
1 
(4) 
(31) 
(31) 
(211) 
(2l To) 
(22) 
(211) 
1 
2 
3 
2 
1 
1 
1 
2 
1 1 
1 
( 22 ) (TTTT) (TIT fob) (II To To) (IT 7T) (mi). 
The essentially distinct graphs with the partitions appertaining to them are 
1111 
111 
21 
11 
1 
1 
11 
(4) 
(31) 
(2l To) 
(22) 
(mu 
(31) 
(22) 
(mi) 
(211) 
(ll IT) 
(211) 
(ITT loo) 
(11 10 10) 
Art. 50. Such graphs are either symmetrical, quasi-symmetrical, or unsymmetrical. 
The symmetrical graphs have three dimensional symmetry, and yield only one 
partition each. 
The quasi-symmetrical have two-dimensional symmetry and yield three partitions 
each. The unsymmetrical yield each six partitions. 
If F ( x ) be the enumerating generating function to a given content, we may write 
F 0) =/i ( x ) + 3/ s (x) + 6/ 3 (a-), 
* The interesting question arises as to the enumeration of the essentially distinct graphs of given 
content. 
