THEORY OF THE PARTITIONS QF NUMBERS. 99 



Art. 22. It is not, I think, immediately obvious that this series can be effectively 

 summed. That this is, in fact, the case appears when the problem is looked at from 

 another point of view. 



I shall show that 



for suppose that the pel-mutation 



,*'/.. .*- a,"~*'a"~* t . 



gives rise to the term cc j *, where (Mi,...*.) is a one-row partition of 2/t. The part to 

 the left of the dividing line may be as small as , and the part to the right as small 

 as a m -i m ; hence 2k has values ranging from 2 to mn-2 ; the partition (Av^i- ,) has 

 parts limited in magnitude to n and in number to m ; and if there were no deductions 

 from the total of partitions the value of L! ( oo , m, n) would be 



but there must be deductions, because every partition of the form rij must be absent ; 

 for the corresponding succession of letters is 



and a dividing line cannot be placed after this succession because every letter prior 

 to ,+! has already appeared to the left of ,+/. Of these omitted partitions there is 

 one, and only one, of each weight, viz. : 



For 



* = 1, ,", ...i", 



i = 2 a^a/as, ai"a" a 3 2 > "i""/"^, 



i = m\ i*a 2 " ... "-!, ... ,"/ ... m ". 



Hence we must subtract 



(mn+1). 



(1) 

 and 



v (n+l)(n+2)...(n+m) (mn+1) 

 Moo.rn.nH* (V)(2);..(m) - ' (1 ) ' 



Since LO = 1, we have also 



(n+l)(n+2)...(n+m) ^ 

 L (oo, ni) n) + L 1 (oo ) m,n)= (i) $)'..(*) (I) 



o 2 



