654 
MAJOR P. A. MACMAHON ON THE 
Qud a given multipartite number, a partition which has q parts and a highest 
figure p is in association with one which has parts and a highest figure q. 
Thus of the multipartite number (13.11.6) we have the partitions 
(333 33l 321 2TT TTO 110 ) 
(664 431 32l) 
derived from the above written graph. 
Art. 47. It is interesting to view the two-dimensional Sylvester-graphs from the 
three-dimensional standpoint. 
Consider the graph 
which, following Sylvester, denotes the unipartite partition (3211) of the unipartite 
number 7. 
In this paper, the graph, read Sylvester-wise in the plane xy and in direction 
Ox, denotes the multipartite number (3211) of content 7. Sylvester’s conjugate 
reading, plane xy and direction Oy, gives the partition (421) but here denotes the 
multipartite number (421). There are four other readings in this theory. The six 
readings are 
Plane xy Direction Ox 
(321]) 
„ xy 
Oy 
(421) 
» w 
Oy 
(421) 
» y z 
Oz (111T 
TTbb l 
„ 
Oz 
(3211) 
yon 
3 3 3 3 
Ox (Ti l 
TTo Too 
q, r) = (3, 1, 4) 
(p, q, r) = (4, 1, 3) 
(p, b r) = (4, 3, 1) 
{p> b r ) = (I) 3, 4) 
{p> b r) = (3, 4, 1) 
{p, q, r) = (I, 4, 3). 
The three multipartite numbers 
(7), 421, (32ll) 
appear each in two partitions. 
In general we establish, in regard to Sylvester-graphs, the six-fold correspondence 
between 
(1) r-partite partitions, containing 1 part and a highest figure p. 
(2) y>-partite partitions, containing 1 part and a highest figure r. 
