THEORY OF THE PARTITIONS OF NUMBERS. 347 



The Functional Equations. 



Art. 2. It is convenient to begin by establishing the functional equations satisfied 

 by the gem-rating function* 



Suppose the lattice to have three unequal rows of a, b, c nodes respectively, and 

 let the part magnitude be restricted by the number I, 



Subject to the parts being in descending order of magnitude in each row and in each 

 column, in every partition each node is either occupied by zero (that is, is unoccupied) 

 or by a number greater than zero and not greater than /. In certain partitions every 

 node is occupied ; such partitions may be constructed by 



(i) Placing a unit at each node, 

 (ii) Superposing every partition enumerated by GF (I 1 ; a, b, c). 



Hence these special, full-based partitions are clearly enumerated by 



Similarly those partitions which are full-based upon a contained lattice specified by 

 the line partition (7/c') are enumerated by 



and we are led to the relation 



GF (I ; a, b, c) = Saf^'GF (/- 1 ; a', 6', </), 



where the summation is in regard to every lattice, specified by (a'b'cf), which is 

 contained in the lattice specified by (abc). 



Art. 3. If from the partitions enumerated by GF (/ ; a, 6, c) we subtract those 

 enumerated by af +t+e GF(J 1 ; a, b, c), we have remaining, in the case of three 

 unequal rows, partitions which include those enumerated by each of the three 

 generating functions 



GF(/;o-l,6,c), GF(/;a,6-l,c), GF(/; a, b, c-1) ; 



and which, by a well-known principle of the combinatory analysis, are enumerated by 

 GF(l;a-l,b,c)+GV(l;a,b-l,c)+G(l;a,b,c-l) 



2 Y 2 



