90 MAJOK P. A. MAOMAHON: MEMOIR ON THE 



deficient Wtice now under consideration is *~L.; hence, if the part magnitude be 



unrestricted, the generating function is 



and if the part magnitude be restricted not to exceed I, 



(l)...(mn-l) 



A simple example, that may be at once verified, is found by taking m = n = 2 and 

 the defective lattice 





 



Here L, = L H = a? and the generating functions are 



1+z 1 



(1) (2) (3) (I) 2 (3) ' 



(1) (2) (3) 

 putting I = 1 we obtain 1 +2a5+x 8 +a; 3 , verified by 



1.11 



. . . . 1 . 1 . 11 



1 x x x 3 a; 3 . 



Now consider the partitions at the nodes of the complete lattice such that one part 

 at least is equal to 1 + 1 and no part exceeds l+l. We obtain all such by placing the 

 part l+l at the node situated at the left-hand top corner and connecting with it all 

 of the partitions at the nodes of the incomplete lattice, which are such that the part 

 magnitude is restricted not to exceed I + 1 in magnitude. 



We thus derive a generating function 



and thence the generating function, which enumerates all partitions at the nodes of 

 the complete lattice, which are such that each has one or more parts at least as great 

 asZ+1, is 



