368 



MAJOR P. A. MACMAHON ON COMBINATORIAL ANALYSIS. 



these say that one which consists in taking 3 from the factor (3), and 1 from the 

 factor (21), we form the first row of our diagram, viz.: 



3 1 



The term resulting from the selected minor operation is 



the operation of D 3 results in four minor operations corresponding to the four ways 

 of picking out a 2 and a 1 from different factors ; we may select the particular 

 minor operation which results in 



and now we add on the second row which denotes this minor operation, and obtain 

 the diagram of two rows. 



We can now only operate in one way with D 3 upon (.) (.) (2) (1) (.) and we finally 

 obtain the diagram of three rows : 



which possesses the property that the sums of the numbers in the successive rows 

 are 4, 3, 3, respectively, while the successive columns involve the partitions (3), (21), 

 (2), (1), (1) respectively. 



The number of such diagrams is A where 



(3)(21)(2)(1)*= . . . +A(43*)+ . . ., ' : 



and A has the analytical expression 



Let us now consider the problem of placing units in the compartments of a 

 lattice of m rows and I columns, not more than one unit in each compartment, 

 in such wise that we can count /x t , /x 2 , . . . p m units in the successive rows, and 

 \i, Xj, . . . X/ units in the successive columns. Take 



As operation. 



As function. 



If 



. . . D M . and 



. . . (I*-) = . . . 



. . . (I*) = A, 



