372 MAJOR P. A. M.uMAHON ON COMBINATORIAL ANALYSIS. 



1 observe that it' k 9, the lattices associated with and enumerated l>y 



include all the row and column-magic squares connected with the natural series ot 

 numbers 1, 2, 3, 4, 5, 6, 7, 8, 9. In general if N = (* + 1), the lattices 

 enumerated by D^ k" s , where k = n 3 , include all the magic squares of order n con- 

 nected with the first n a numbers. If we could further impress the condition that no 

 compartment number is to be twice repeated, we would be successful in enumerating 

 the magic squares divorced from the diagonal property. This seems to be a matter 

 of difficulty, which is increased if an attempt be made to introduce diagonal and 

 other conditions to which certain classes of magic squares are subject. 



It may be gathered from what has been said, that every case of symmetric function 

 multiplication is connected with a theory of lattice combinations. For if we take 

 as function 



and as operation 



we have 



D,J) 7 D, . . . (X^^ . . . ((Xo/i 3 i>., ...)(...)... fatM . . .) = A, 

 where 



that is to say, we multiply together a number of monomial symmetric functions so as 

 to exhibit it as a sum of monomial functions ; in this sum we find a particular 

 monomial function affected with a numerical coefficient A which, as shown by the 

 present theory, is the number which enumerates lattices of a certain class easily 

 definable. Thus, in the present instance, if the partition (pqr . . .) involve t parts, 

 the lattices have s columns and t rows ; the operation ~D f acts, through its various 

 partitions, upon the product of monomials, and any mode of picking out a partition 

 of p from the factors of the product, one part from each factor, constitutes a minor 

 operation which yields the first row of a lattice ; the operation D ? is similarly respon- 

 sible for all the second rows of the lattices, and finally every resulting lattice possesses 

 a property which may be defined as under : 



The numbers in the successive rows are partitions of the numbers p, q, r, . . . 

 respectively, and in the successive columns are the partitions (X^^ . . . ), 

 (^a^a ...),... (X,JU,,K, . ) respectively. Such are the lattices enumerated by 

 the number A. One is reminded somewhat of CAYLEY'S well-known algorithm for 

 symmetric function multiplication (invented by him for use in his researches in the 



