168 MA JOE P. A. MAcMAHON: MEMOIR ON THE 



Consider in general the arrangement 



where the a pairs are in order and the /3 pairs out of order. 

 For this particular arrangement the enumeration is given by 



Put 



u z = 1 + X, 



The generating function to be considered is 



1 + X+Y + XY + YX + XYX + YXY+XYXY+YXYX+.... 



The X, Y in a product occurring alternately in all possible ways. 

 This function is 



1-XY 



or 



or 



xn x yu y 



x-y 

 since 



xu x a u x +l = yUy 2 -Vy+I = 0. 

 Now 



x ~~y 



and the coefficient of x"~*?/* in the function is none other than 



(aa ; s). 

 Hence 



(aa ; s) = (aa ;), 

 a remarkable theorem. 



21. I will now obtain the generating function for the numbers (ab ;). Since 



, 

 a a+l 



& Wi / -ii(' 

 \ a / o i \ a 



