and then we may evidently regard it as a unipartite partition on the points of a lattice,
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
α1 ≥ α2 ≥ α3 ≥ ... ... αn
α1 + α2 + α3 + ... + αn = n,
the points being in a straight line. In the simplest example of the three-dimensional graph we have to solve the system
| α1 ≥ α2 | ||
| ≚ ≙ | α1 + α2 + α3 + α4 = n, | |
| α3 ≥ α4 |
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 - xs = (s), the generating function is given by the product of the factors
| x | ||||
| (l + 1) | . | (l + 2) | .... | (l + m) |
| (1) | (2) | (m) | ||
| (l + 2) | . | (l + 3) | .... | (l + m + 1) |
| (2) | (3) | (m + 1) | ||
| . | . | .... | . | |
| . | . | .... | . | |
| . | . | .... | . | |
| (l + n) | . | (l + n + 1) | .... | (l + m + n - 1) |
| (n) | (n + 1) | (m + n - 1) | ||
| y |
one factor appearing at each point of the lattice.