THEORY OF THE PARTITIONS OF NUMBERS. 89 



Assume the truth of the theorem in the case of 



for all values of I ; then 



Putting x l = 6, 

 , + 





and, therefore, 



)_ fa 





(l).-.(n-l) 



and thence 



Hence 



by induction. 



To generalize this method, I take a lattice which is complete but for the node at 

 the left-hand top corner 



and first determine the generating function for partitions such that the descending 

 order of part magnitude is in evidence in each row and in each column. I take the 

 number of rows to be m, and the number of columns n. A slight consideration shows 

 that if L, be the sub-lattice function of order * for the complete lattice, that of the 

 VOL. ccx r. A. N 



