8 MAJOR P. A. MAcMAHON: MEMOIR ON THE 



whert- x < t and the product II has reference to every pair of numbers a t drawn 

 from the succession a,, a,, ..., a m . 



This result will be found to furnish an important key to the solution of the 

 problems l)efore us. 



It IB possible, by the method employed, to consider the generating functions for 

 partitions at the nodes of an incomplete lattice, and I shall use GF (/; a, b, c, ...) to 

 denote that which has reference to a lattice whose successive rows involve a, b, c, ... 

 nodes, respectively, the part magnitude being restricted by the number I. In this 

 notation GF (/, m, n) may alternately be written GF (1; n m ) or GF (I ; in"), wherein n m 

 will denote m rows each of n nodes. 



I derive from every lattice, complete or incomplete, a lattice-function of x, a,nd this 

 function depends, like the generating function, not only upon the specification of the 

 lattice, but also upon the number / which limits the part magnitude. I denote this 

 function by L (/, m, n) or by L (I; a, b, c, ..), according as the lattice is complete or 

 incomplete. In cases where no confusion can arise, I simply write L for brevity. 

 Art. 5. I will now explain the formation of the functions 



L(oo, m, n) and L ( oo ; a, b, c, ...) ; 

 and then establish the fundamental propositions 



L ( , in. n) 



= 



GF ( oo ; , b, c, ...) = 



In the next place I will explain the formation of the functions 



L(l,m,n) and L (/; a, b, c, ...), 

 and establish the fundamental propositions 



GF (I, m, n) = Lfe L_ 



' 



Art 6. Consider an incomplete lattice having 3, 2, 1 nodes in the rows respectively, 

 d-ttenmt mteger* (say the first six) be placed in any manner at the 

 ch w,se that descending order of magnitude is in evidence b each row and 

 column ; such an arrangement may be 



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