364 MAJOR P. A. MACMAHON OX THE THEORY OF PARTITIOXS OF NUMBERS. 
the partition 4 3 3 3 1 has the mixed graph 
11111 
2 2 2 2 
1 
as well as its ordinary unit-graph. 
In one case the mixed graph is composed entirely of units, and is, moreover, the 
graph conjugate to the unit graph. 
This happens when 
(P., P,, P 3 , . . .) = (1, 2 , 3, . . .). 
Thus, q\id these elements, 
4 3 3 3 1 
has the mixed (here the conjugate) graph 
11111 
1111 
1111 ’ 
1 
Art. 77. Observe that a partition may be such qvd the parts which actually appear 
In ii, or it may be cpid, in addition, certain parts which might appear, but which 
happen to be absent. A mixed graph corresponds to each such supposition. 
Ex. gr, :— 
Partition. 
Qua elements. 
Graph. 
4 3 
. „ 
4, 3 
3 3 
1 
4 3 
4, 3, 1 
1 1 
2 2 
1 
4 3 
4, 3, 2 
2 2 
1 1 
1 
4 3 
4, 3, 2. 1 
1 1 
1 1 
1 1 
1 
We thus arrive at a generalization of the notion of a conjugate partition, and are 
convinced that the proper representation of a Ferrers-graph is not l)y nodes or points, 
but by units. 
