" 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.--

3 3 3 3 2 2
3 2 2 1
3 2 1

and then we may evidently regard it as a unipartite partition on the
points of a lattice,

0 +-----+-----+-----+-----+------- x
| | | | |
| | | | |
+-----+-----+-----+-----+-------
| | | | |
| | | | |
+-----+-----+-----+-----+-------
| | | | |
| | | | |
+-----+-----+-----+-----+-------
| | | | |
y

the descending order of magnitude of part being maintained along _every_
line of route which proceeds from the origin in the positive directions
of the axes.

This brings in view the modern notion of a partition, which has
enormously enlarged the scope of the theory. We consider any number of
points _in plano_ or _in solido_ connected (or not) by lines in pairs in
any desired manner and fix upon any condition, such as is implied by the
symbols >=, >, =, <=, <>, as affecting any pair of points so connected.
Thus in ordinary unipartite partition we have to solve in integers such
a system as

[a]1 >= [a]2 >= [a]3 >= ... [a]n

[a]1 + [a]2 + [a]3 + ... + [a]n = n,
([a] = [alpha])

the points being in a straight line. In the simplest example of the
three-dimensional graph we have to solve the system

[a]1 >= [a]2
v = [a]1 + [a]2 + [a]3 + [a]4 = n,
= v
[a]3 >= [a]4

and a system for the general lattice constructed upon the same
principle. The system has been discussed by MacMahon, _Phil. Trans._
vol. clxxxvii. A, 1896, pp. 619-673, with the conclusion that if the
numbers of nodes along the axes of x, y, z be limited not to exceed the
numbers m, n, l respectively, then writing for brevity 1 - x^s = (s),
the generating function is given by the product of the factors

+----------------------------------------------x
|
| (l + 1) (l + 2) (l + m)
| ------- . ------- ... -------
| (1) (2)