82 MAJOR P. A. MACMAHON: MEMOIR ON THE 



Every one of the 



arrangements of the different integers at the nodes of the lattice will thus have a 

 power of x associated with it, and taking the sum of them all I obtain the lattice 



function ) _ 2^+*+*+... 



L ( oo ; a lf a 3 , a tt ...) z*& 



Art. 7. I will set out at length the formation of L ( oo ; 3, 2, 1). 



* 



X* 



a, 



, aa/3y|a/3, aa/3/3y|a, 



a;* 



a; 



a, a/ 8 1 aay | y8, a)8 1 a | ay, 

 x 7 a; 8 



so that 



L(oo; 3,2,1)= 



It is obvious that this process can be carried out in respect of any lattice, complete 

 or incomplete, and that the number of different Greek letters involved will be equal 

 to the number of rows. 



Art. 8. To show the connexion between such a lattice function and the corre- 

 sponding generating of partitions at the nodes of the lattice I proceed as follows : 

 We have to establish the relation 



As the simplest possible case (with a trivial exception) consider the complete lattice 

 of 2 rows and 2 columns 



and any numbers, equal or unequal, to be placed at the four nodes in such wise that 



