THKORY OF THE PARTITIONS OF NIMH! I, 's 77 



I counted, &s far as weight 16, the numbers of the partitions by separately 

 counting those whose graphs possess one, three, and six readings. At weight 13 a 

 discrepancy appeared, because of that weight there are only two graphs which have 

 one reading, and, on the assumption that the remaining graphs could be read in either 

 three or six ways, it was clear that the number of the partitions must l>e = 2 iriod 3 ; 

 but the coefficients of a- 13 in the supposed generating function was found to be 2485, 

 which is = 1 mod 3. It thus became clear either that the reasoning from the graphs 

 was wrong, or that the generating function was at fault. The discrepancy was 

 cleared up by the discovery that at weight 13 graphs with two readings present 

 themselves for the first time. The simplest of these is 



331 



211 of weight 13. 



2 



The property possessed by these partitions is that the successive rows are the 



conjugates of the successive columns without being identical with them ; that is to 



say, that the successive rows are not to be self-conjugate partitions. Thus, 331, 



211, 2 are conjugates of 322, 31, 11 respectively. The reading of the corresponding 



hree-dimensional graph in the six modes gives either 



331 322 



211 or 31 

 2 11. 



The separate enumeration of these forms is a matter for future enquiry. 



Art. 1. Turning now to the substance of this communication, I shall introduce a 

 new plan of procedure which is applicable when the places for the parts of the 

 partitions are given by the nodes of two-dimensional lattices, which may be complete 

 or incomplete. In every case I suppose the part magnitude to be not greater than /, 

 and when the lattice is complete, I suppose it to have m rows and n columns. The 

 generating function which gives by the coefficients of x v the number of the partitions 

 of w of the nature considered will be denoted by GF (I, m, n). 



In the excellent notation of CAYLEY and SYLVESTKR I shall denote the algebraic 

 expression 1 x' by (s), employing Clarendon type for the letter s, and thus 1 x 

 by (1) and 1 x l+1 by (l+l), using always the Clarendon type in order to differentiate 

 such notation from that in which between the brackets the ordinary Roman type is 

 employed ; the latter will, in general, denote integers *, 1, l+l, as the case may be. 

 The notation is perfect for the purpose in hand, because it merely exhibits and 

 concentrates attention upon the exponent *, which is the essential part of the 

 expression, and the only part that in many cases it is necessary to handle 

 algebraically. Further, in several instances, identities involving such expressions in 



