THEORY OF THE PARTITIONS OF NUMBERS. 167 



the last term of the series being the square of 



a\ c 2. a! 



or of 



ascending as a is even or uneven. 



The permutations enumerated by (aa ;) are those of 2a numbers 



all different and subject to a + 2 descending orders corresponding to the a columns 

 and the 2 rows. 



20. Consider now other permutations such that whilst the row numbers are in 

 descending order, exactly s of the a column pairs are not in descending order. 



Let (aa ; s) be the number of such permutations. I propose to show that 



(aa ; .s-) = (aa ; 0) = (aa ;) 



for all values of s, from ,<? = to ,s = a. 

 Ex. gr., the permutations enumerated by 



(22 ; 0) are 43 42 

 21 31, 



(22 ; 1) are 41 32 



32 41, 



(22 ; 2) are 31 21 

 42 43. 

 To establish this theorem, since 



and 



xu z 2 = u z l, 

 we find 



(aa;) = (a-1, a-l;) + (a-2, a-2 ;) (11 ;)+ ... + (11 ;) (a-2, a-2;) + (a-l, a-1;). 



The right-hand side of this identity is equal to 



(aa; I), 



for it consists of a terms, of which the first enumerates the permutations in which the 

 pair m^n^ is out of order, the second those in which the pair m 3 n a is out of order, and 



so on. Hence 



(aa;) = (aa ; 1). 



