Again, if we restrict the part magnitude to i, the largest angle of nodes contains at most 2i - 1 nodes, and based upon a square of θ² nodes we have partitions enumerated by the coefficient of aθxn in the product (1 + ax) (1 + ax3) (1 + ax5) ... (1 + ax2i-1); moreover the same number enumerates the partition of ½(n - θ²) into θ or fewer parts, of which the largest part is equal to or less than i - θ, and is thus given by the coefficient of x½(n-θ²) in the expansion of
| 1 - xi-θ+1. 1 - xi-θ+2. 1 - xi-θ+3. ... 1 - xi | , |
| 1 - x. 1 - x2. 1 - x3. ... 1 - xθ |
or of xn in
| 1 - x2i-2θ+2. 1 - x2i-2θ+4. ... 1 - x2i | xθ2; |
| 1 - x2. 1 - x4. 1 - x6. ... 1 - x2θ |
hence the expansion
| (1 + ax) (1 + ax3) (1 + ax5) ... (1 + ax2i-1) = 1 + Σ θ=iθ=1 | 1 - x2i-2θ+2. 1 - x2i-2θ+4. ... 1 + x2i | xθ² aθ. |
| 1 - x2. 1 - x4. 1 - x6. ... 1 - x2θ |
There is no difficulty in extending the graphical method to three Extension to three dimensions. dimensions, and we have then a theory of a special kind of partition of multipartite numbers. Of such kind is the partition
(a1a2a3..., (b1b2b3..., (c1c2c3..., ...)
of the multipartite number