THEORY OF THE PARTITIONS OF NUMBERS. 165 



if c be the smallest number involved, the arrangements are of two types, viz., 



c 



or 



6 



except when the rows contain the same number of nodes ; then there is the one type 



e 



Hence, a moment's consideration establishes the relations 



(a, 6-1;) 



when a > b, and 



(aa;) = (a, a- 1 ;). 



Treating these as difference equations it is easy to obtain the result 



> (a + M! / 7 , i\ /a + b\ab+l 



;) = /-* - \-r-n ( a -v + l ) = \ - r~ > 



(a+1)! 6P \ a / a+1 



, x (2) ! /2a\ 



(; '~a+i!! = 



18. This case of two rows is worth a special examination before proceeding to a 

 greater number of rows. First consider the generating function of the numbers (aa ;) : 



u x = 1(aa;}x a = I 

 If we expand 



( 



we find that the general term after the first is 



and thence 



^ = 



and 



xUf 



exhibiting a remarkable property of u x . 

 Reverting to the difference equation 



= 0, 



(aa;) = (a, a-1;) 



= (a, a-2 ;) + (a-l, a 1 ;) 



= (a, a-3;) + 2(a-l, a-2;) 



= (a, a-4 ;) + 3 (a-1, a-3 ;) + 2 (a-2, a-2), 



