THEORY OF THE PARTITIONS OF NUMBERS. 91 



ami it is easy to verify that this expression added to the expression already found for 

 GF (I, m, n) is, in fact, equal to 



that is, to GF( oo, m, n). 



Art. 15. This main proposition involves the whole theory of the partitions at the 

 nodes of an incomplete lattice ; it gives the true generating function without 

 redundant terms, and this only needs examination and, where possible, simplification. 

 Such simplification is apparently always possible when the lattice is complete. 

 Moreover, there is the task of exhibiting L ( ; a,, a a , 3 , ...) as a product of outer and 

 inner lattice functions and of finding the algebraic expression of L, ( oo ; a,, a,, a s , ...). 

 There is an important and quite general property of the lattice function which must 

 noW be explained. If a lattice be read by columns instead of by rows its specification 

 changes from a partition to the conjugate partition, and it is a trivial remark that the 

 generating function of partitions at the nodes is not altered. In fact, if the rows 

 possess a,, a.,, ..., a m nodes and the columns 6,, & ..., b n nodes 



GF(Z ; a,, a* ..., a m ) = GF(/ ; 6,, 6, ..... &). 



Moreover, since the generating function is the quotient of the lattice function by 

 an algebraic function which depends merely upon the number of nodes, it is clear 



that 



L( oo ; a lf a,, ..., a m ) = L( oo ; 6,, 6, ..... 6.), 



Oj, a a ..... a m ) = L(l; 6,, 6, ..... b n ), 



(a,, a,, ..., a m ) and (& &.,,..., &) being conjugate partitions. 

 From the last written relation we find 



(l+l) ... (l+,5a) + (l) ... (l+^a-l)M oo ; a,, a,, ..., a.) 



+(l-l) ...(l+-5a-2)L,( oo ;o,, a,, ...,)+... 



= (l+l) ... (l+3a) + (l) -.. (l+5a-l)L, ( oo ; 6,, &,, ..., 6.) 



+ (l-l) ... (l+^a-2) L( co : &i. 6*. 6 )+- 

 Putting herein I = 1, 2, ... in succession, we establish that 



L,( ; i, s, ><*.) = L.(oo ; fc,,& . ..,&), 



thence 



L. (I ; en, o,, ..., ,) = L. (/ ; &,, & .... &), 



proving that the sub-lattice functions also do not change in passing from a lattice fa 



the conjxigate lattice. 



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