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IX. Memoir on the Tkeori/ of the Partitions of Numbers. Part VI. Partitions 



in Two-dimensional Space, to which is added an Adumbration of the 



Theory of tiie Partitions in Three-dimensional Space. 



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



Received June 13, Read June 29, 1911. 



Introduction. 



I RESUME the subject of Part V.* of this Memoir by inquiring further into the 

 generating function of the partitions of a number when the parts are placed at the 

 nodes of an incomplete lattice, viz., of a lattice which is regular but made up of 

 unequal rows. Such a lattice is the graph of the line partition of a number. In 

 Part V. I arrived at the expression of the generating function in respect of a two- 

 row lattice when the past magnitude is unrestricted. This was given in Art. 16 in 

 the form 



GF ( oo 6) = (!)+**' (-b) 



I remind the reader that the determination of the generating function, when the 

 part magnitude is unrestricted, depends upon the determination of the associated 

 lattice function (see Art. 5, loc. cit.). This function is assumed to be the product of 

 an expression of known form and of another function which I termed the inner lattice 

 function (see Art. 10, loc. cit.), and it is on the form of this function that the interest 

 of the investigation in large measure depends. All that is known about it a priori 

 is its numerical value when x is put equal to unity (Art. 10, loc. cit.). The lattice 

 function was also exhibited as a sum of sub-lattice functions, and it was shown that 

 the generating function, when the part magnitude is restricted, may be expressed as 

 a linear function of them. These sub-lattice functions are intrinsically interesting, 

 but it will be shown in what follows that they are not of vital importance to the 

 investigation. In fact, the difficulty of constructing them has been turned by the 



* 'Phil. Trans.,' A, vol. 211, 1911. 

 VOL. GXIXI. A 479. 2 Y 9.9.11 



