III. Memoir on the Theory of the Partitions of Numbers. Part V. Partitions 



in Tioo- dimensional Space. 



By Major P. A. MACMAHON, R.A., D.Sc., F.It.S. 



Received December 31, 1910, Read January 26, 1911. 



Introduction. 



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. 



Before proceeding to explain the new method of research which enables this paper 

 to make a notable advance, I must hasten to correct an error which I had not 

 detected at the time a former paper was written. 



It will be remembered that partitions in a plane are such that there is a graphical 



* "Memoir on the Theory of the Partitions of Numbers," 'Phil. Trans. Roy. Soc.,' A, 1896, vol. 187, 

 pp. 619-673; 1899, vol. 192, pp. 361-401 ; 1905, vol. 205, pp. 37-59. 



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