THEORY OF THE PARTITIONS OF NUMBERS. 79 



;md tin- otln-r is that given in Part II. of this Memoir, and also hy FOBSYTH, 



Tliis generating function may be regarded as enumerating partitions 



(i) At the nodes of a lattice of 2 rows and n columns (or of n rows and 

 2 columns) 



the part magnitude l>eiiig unrestricted ; 



(ii) At the nodes of a lattice which has the number of unrestricted, 



columns 



the number of equal to n and the part magnitude restricted to 



be > 2. 



Tin- found result shows that the number of partitions of w is equal to the number 

 of ways of composing w with 



one kind of unity, 



two kinds of twos, 

 two kinds of threes, 



two kinds of u's, 

 one kind of n+ I , 



but all attempts to establish a one-to-one correspondence have failed. Had this 

 proved to have been feasible it miyht have been extended to prove the similar results 

 for GF ( oo, m, n) where in > 2. 



Art. 4. The linear Diophantine Analysis, which was applied to the same question 

 in an earlier part of this Memoir, having also failed to establish general results, 

 recourse has been had to a plan suggested by Part IV. of the Memoir,* and a 

 considerable advance has been made. In that paper I considered the number of 

 different ways in which k different numbers can lie placed at the nodes of a lattice, 

 complete or incomplete, the number of nodes being A\ and the numbers being placed 

 in such wise that descending order of magnitude is in evidence in each row from West 

 to East and in each column from North to South. 



In the paper quoted I showed that if the rows involve a,, u 3 , ..., a m nodes 

 r. 'spectively, where, of* course, a^a.^ ... S a m , the number of ways of arranging the 

 S different numbers at the nodes is 



(a^n-1)! (a,+m-2)\ ... (o.-. + l)! a,!,"' 

 * "Memoir on the Theory of the Partitions of Numbers," ' Phil. Trans.,' A, 1908, voL 209, pp. 153-175. 



