THEOKY OF THE PARTITIONS OF NUMBERS. 169 



As yet we have assigned no meaning to (ab ;) when a < b, but retaining such terms 

 and adding to the term 



ry 



terms given by placing a equal to zero, we obtain the suggestive redundant 

 generating function 



X 



\-x-y 

 in the expansion of which we require only those terms involving 



scY 



in which a 2t 1). 



To eliminate the terms containing x~ l we have merely to add 



y 



X , 



and then we obtain 



l-2y 



l-y, l-x-y 



We might also seek to remove those terms in x a t/' for which a < b, but for my 

 present purpose the redundant form is quite as convenient and infinitely more 

 suggestive. * 



22. The result 



/a + b\a b+l 

 = a 



leads to the observation that the number obtained is precisely that obtained in 

 Section 1 for the number of arrangements of the letters in 



a." 



such that drawing a line between any two letters the number of a's to the left of the 

 line 2: to the number of )S's to the left of the line. Also that 



is the solution of the probability question for a + b electors. 



The one-to-one correspondence is easily established, for suppose 



* 



86531 



742 



I have found that the reduced generating function is 



ix y 

 VOL. CCIX. A. Z 



