gg MAJOR P. A. MACMAHON: MEMOIR ON THE 



we obtain a part of the generating function 



Thence 



. 

 GF(/,3,2) = (!)() ...(6) 



L(/. 3.2) 

 -(!)() ...(6)' 



In general, when the lattice has 2a nodes, we have a set of inequalities belonging 

 to L,( oo ; a,, a 3 , ...) which give rise to the generating function 



s) , 

 L,(oo ; j,a 3 , a 3 , ...); 



(1) (8) ...(*) 



and thus the above-given fundamental relations are established. 



Art. 14. The generating functions for two-dimensional partitions GF (I, m, n) has 

 been found in terms of lattice functions in the form 



(l+l) ... (l+mn) L.+ (I) ... (l+mn-1) L, + ...... + (l-^+l) ... (l-^+mn) Iy 



(l)...(mn) 



If we subtract these partitions from those enumerated by GF ( oo, m, n), we are left 

 with those partitions which contain one part at least equal to or greater than l+l. 

 I shall show how to determine directly the generating function for these in terms of 

 lattice functions. To lead up to the proof, I will give an inductive proof of the 

 theorem 



Taking the parts at n nodes in one row 



the partitions which have a highest part equal to l+l will be obtained by placing the 

 part l+l to the left of each of the partitions enumerated by GF (l+l, 1, ?i-l). 



Hence the whole of the partitions which have one part at least' equal or greater 

 than l+l are enumerated by 



and 



,u) = GF(oo, 



