IX. Memoir on the theory of the partitions of numbers. - Part VI. Partitions in two-dimensional space, to which is added an adumbration of the theory of the partitions in three-dimensional space

Abstract

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 ( ∞ ; a, b ) = (1) + x b +1 (a - b) / (1) (2) ... (a+1). (1) (2) ... (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 à 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, hut 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 formation and solution of certain functional equations which lead in the first place to the required generating functions, and in the second place to an exhibition of the forms of the sub-lattice functions. To previous definitions I here add the definition of the inner lattice function when there is a restriction upon the part magnitude, and it will be shown that the generating, lattice, and inner lattice functions satisfy certain functional equations both when there is not and when there is a restriction upon the part magnitude.