THEORY OF THE PARTITIONS OF NUMBERS. 171 



/ 



we find the analytical results 



(a-l,a;) = 0, 



(6-1, + !;) = -(&;), 



and the latter of these is not so far interpretable. 

 Passing to three rows 



containing a, b, and c nodes respectively, it is seen that the smallest of the 

 different numbers must be situated at the right-hand nodes of some row, unless such 

 row contains as many nodes as the row beneath. 

 Hence the difference equation 



(abci) = (ft- 1, b, c;) + (a, b-l, c ;) + (ft, b, c-1 ;), 

 provided that (abc ;) = when either 



ab+l = or bc+1 = 0. 

 I find such a solution of the difference equation to be 



This is only interpretable when 



a > b > c , 

 but analytically 



+ (abc;) = +(b-l,c-l,a + 2;)= +(c-2, a+ 1, b+ 1 ;) 

 = -(a, c-1, 6+1;)= -(&-!, a+l,c;) = -(c-2, b, a + 2 ;), 



relations which are useful for the manipulation of the functions. 

 The sum 



with the inclusion of redundant terms I find to be 



x y x 



l-x-y-z 



an expression which is remarkably suggestive. 



z 2 



