(a1 + b1 + c1 + ..., a2 + b2 + c2 + ..., a3 + b3 + c3 + ..., ...)

if

a1 ≥ a2 ≥ a3 ≥ ...; b1 ≥ b2 ≥ b3 ≥ ..., ...
a3 ≥ b3 ≥ c3 ≥ ...,

for then the graphs of the parts a1a2a3..., b1b2b3..., ... are superposable, and we have what we may term a regular graph in three dimensions. Thus the partition (643, 632, 411) of the multipartite (16, 8, 6) leads to the graph

and every such graph is readable in six ways, the axis of z being perpendicular to the plane of the paper.

Ex. Gr.

Plane parallel toxy,directionOxreads(643, 632, 411)
 ”   ”xy,  ”Oy ”(333211, 332111, 311100)
 ”   ”yz,  ”Oy ”(333, 331, 321, 211, 110, 110)
 ”   ”yz,  ”Oz ”(333, 322, 321, 310, 200, 200)
 ”   ”zx,  ”Oz ”(333322, 322100, 321000)
 ”   ”zx,  ”Ox ”(664, 431, 321)

the partitions having reference to the multipartite numbers 16, 8, 6, 976422, 13, 11, 6, which are brought into relation through the medium of the graph. The graph in question is more conveniently represented by a numbered diagram, viz.—

333322
3221
321