IIIF.ORY OF THE PAUTITIONS OF NUMBERS. 107 



The relations between the functions L, and L,,_, yield remarkahle algebraical 

 identities. 



So far I have established a new method for the discussion of these questions of the 

 a mil laments of numbers, and have made some progress with the simplification of the 

 fundamental expression arrived at for the generating function. I have further shown 

 the great probability of the outer lattice function being the whole lattice function 

 whenever the lattice is complete. Before this can be rigidly established, I believe 

 that a further study of the theory of the incomplete lattice will be necessary. From 

 many particular incomplete lattices that have already been worked out this 

 investigation promises well, and I hope in due course to lay the results before the 

 Society. 



POSTSCRIPT. 



There is an analogous theory which is concerned with the totality of the permuta- 

 tions of a^'a/ 3 . . . a/' We thus obtain permutation functions which possess elegant 

 properties. The functions also arise from the theory of partitions. 



Suppose that we desire the number of two-dimensional partitions of a number such 

 that the nodes of the lattice descending order of part magnitude is in evidence in 

 each row but not necessarily in each column. It is immediately evident that the 

 generating function of such at the nodes of a lattice which contains p } , p t , ...,p n nodes 



in the successive rows is 



1 



(l)...( Pl )(l)...( P2 ) (!)...(*)' 



whether the numbers p lt p a , ..., p, be in descending order of magnitude or not. This 

 fact enables us to determine the lattice function and the sub-lattice functions 

 derivable from the whole of the permutations of the letters af'ct* . . . of when we 

 may suppose the exponents p lt p 3 , .... p H to be in descending order of magnitude and 

 establishes also that these functions are invariant for any permutation of 'p lt p t , ...,p n 

 in the product a/'a/ 1 . . . /. 



We may proceed in exactly the same manner as when the restricted permutations 

 were under view. Taking the lattice corresponding to a l 3 a. t 3 a 3 and arranging 6 

 different numbers in any way so that descending order is in evidence in the rows 



3 2 1 



6 5 a a aja 3 [ iii 



4 



we have the arrangement figured and the corresponding Greek-letter succession, 

 yielding a portion 



(lTl6) . H , 



of the generating function. 



P 3 



