THEORY OP THE PARTITIONS OP NUMBERS. s;i 



there is descending order of magnitude in both rows and in both columns ; say that 

 the numbers are 



P <1 



r s 



subject to the conditions p ^ q 2: s, p ^ r i s. 

 It is clear that we must either have 



(i) p ^. q i r S: s or (ii) p i r > q 2s s ; 



and that these two systems do not overlap. 



If (i) obtains we may perform the summation ^x p+9 * r+ ' by writing r = s + A., 

 q = s + A + B, p = s + A + B + C, where A, B, C are arbitrary positive integers, zero 



included ; the sum is thus 



2z CH 



and since C, B, A, s may each of them assume all values ranging from zero to infinity, 

 the sum is clearly 



if, on the other hand, the parts of the partition have such values that (ii) obtains, we 



may write 



7 = s+A, r = s + A+B+l, p = s+A+B+C + 1, 



and we have the sum 

 which is equal to 



By addition we have 



GF ( oo ; 2, 2) = 



and it will be noted that 1+z 8 = L( oo ; 2, 2), derived, as above, from the lattice 



arrangements 



43 42 



21 31 



The fact is that the alternatives (i) and (ii) exist because there are two lattice 

 arrangements of unequal numbers, and the signs of equality and inequality are 

 arranged in (i) and (ii) so that the required sum may be separated into two non- 

 overlapping systems in correspondence with the lattice arrangements. The fact that 



M 2 



