366 MAJOR P. A. MAcMAHON ON COMBINATORIAL ANALYSIS. 



Since D, (1)" = ( )(1)"~*', D Ti may be regarded as breaking up into ( ) minor 

 \*i/ \*v 



operations, each of which consists in erasing TTJ of the n factors 



(1)(1)(1). - - (1), 

 and replacing them by units. On the diagram we denote one such minor operation 



Keeping to this term, 



1.1.(1).1.(1). 1.1. (.!)(!). . . 



and operating with D rj , we find the operation breaking up into ( l } minor opera- 



\ ""s r 



tions, each of which consists in erasing 7r 2 of the factors, and replacing them by units. 

 Selecting one of these minor operations, we find that the corresponding term has been 

 obtained by operations conforming to the diagram of two rows 



n of units -V, 

 n of units -Tt t 



Proceeding in this manner, we finally arrive at a lattice of n rows and n columns, 

 such that there is one and only one unit in each column, while the numbers of units 

 in the 1st, 2nd . . . rath rows are TT I} tr z , . . . ir n respectively. 



We have thus the associated problem on the lattice, and we obtain 



TT l ! 7T, ! ... IT, ! 



representations on the lattice corresponding to the permutations of the quantities in 



a/". 



The simple introductory examples lead one to expect that the method will be found 

 capable of dealing with questions either of a chess board character or which are con- 

 cerned with rectangular lattices. Further, the idea of lattice rotation gives promise 

 of leading to theorems of algebraic reciprocity and of reciprocity in the theory of 

 numbers, features almost inseparable from any lattice theory. 



Art. 3. The graph of a partition. It is convenient to have before us the connexion 

 between this theory and the SYLVESTER-FERRERS graph of a unipartite partition. 



Consider the operation T> 3 DD l and the symmetric function (1*) (I 3 ) (1), (32 3 1) 

 being the partition conjugate to (431). 



DgD/Dj (I 4 ) (1 s ) (1) = D/D! (1 s ) (l), the operation D 3 erasing one part, viz. : 

 unity from each factor ; this we denote as usual by 



HUD; 



