III. Memoir on the theory of the partitions of numbers. —Part V. Partitions in two-dimensional space

In previous papers I have broached the question of the two-dimensional partitions of numbers—or, say, the partitions in a plane—without, however, having succeeded in establishing certain conjectured formulas of enumeration. The parts of such partitions are placed at the nodes of a complete, or of an incomplete, lattice in two dimensions, in such wise that descending order of magnitude is in evidence in each horizontal row of nodes and in each vertical column. No decided advance was made in regard to the complete lattice, and the question of the incomplete lattice is considered for the first time in the present paper. I return to the subject because I am now able to throw a considerable amount of fresh light upon the problem, and have succeeded in overcoming most of the difficulties which surround it. In fact, I am now able to show how the generating functions may be constructed in respect of any lattice, complete or incomplete, in forms which are free from redundant terms. I have not succeeded, so far, in giving a general algebraic expression to the functions, but, in the case of the complete lattice, I have shown that an assumption as to form, consistent with all results that have been arrived at in particular cases, leads at once to the expression that has been for so long the conjectured result. For the complete lattice of two rows, and for the incomplete lattice of two rows, the results have been obtained without any assumption in regard to form, and must be regarded as rigidly established.