THEORY OF THE PARTITIONS OF NUMBERS. 



51 



Art. 140. To find the number of ground "general magic squares" corresponding to 

 the sum unity, we have merely to pick out the coefficient of X/u, ; we thus find 



even order 2n number is 8 ( ) a- a n ~ 3 <ri a , 



w 



uneven order 2n+l number is 8 (2) tr 2 n ~ 2 cr 1 3 +o- 2 ", 



wherein it must be remembered that the a- products are to be expanded in powers 

 of s and then s m put equal to m I 



In the general results the coefficient of 



xy* 



gives the number of squares in which the row and column sums .are unity and the 

 dexter and sinister diagonals' sums are I, m respectively. 

 I give the following table of values of simple a- products : 



0-2 



r. 



_ 2_ 2 

 CTj <T 2 



(TjOj 





The numbers cr/ = (s l) p denote the number of permutations of p letters in which 

 each letter is displaced and constitute a well-known series. 



The remaining numbers are' readily calculated from these by the formula 



O-/G-/ = o-/ +2 o-/- 1 -2cr/ +1 o-/- 1 -<o-/- 1 . 



Art. 141. Another solution of the same problem yielding a more detailed result is 

 now given. 



For the even order 2n I directly determine the coefficient of 



in the product above set forth. 



H 2 



