378 MAJOR P. A. MACMAHON ON COMBINATORIAL ANALYSIS. 



Compare the difference formula 



Au,v,w, = (EE'E" l)t*,v.,?0., 



= (A + A' + A" + A'A" + A"A + AA' + AA'A") u f v f w fi 



where u x is only operated upon by E and A, v x by E' and A', w f by E" and A". 



Art. 9. Consider the lattice theory connected with the operation D and zero- 

 part partition functions. 



Take the function 



(0*) (0") (0") . . . 



X, /*, v, . . . being in descending order ; if it be multiplied out, it will appear as a 

 linear function of (O x ), (O x + 1 ), . . . (o* + * + r+ ), the coefficients being positive (cf. 

 Second Memoir, loc. cit., p. 102). 



To find therein the coefficient of the term (0') we must operate with D ', and the 

 sought coefficient is the resulting numerical term. If the factors (0 X )(0 M )(0 V ) ... be 

 t in number, we are concerned with lattices of t columns and s rows. The first opera- 

 tion of D results in a first row whose compartments contain t or fewer zeros placed 

 in any manner so that not more than one zero is in each compartment ; similarly, for 

 the successive rows and the final lattice is subject to the single condition that the 

 numbers of zeros in the successive columns are X, ft, v, . . . respectively. The number 

 of such lattices is 





or, symbolically, (P - I}' () (?) (')... 



We have thus the analytical solution of a distribution problem upon a lattice. 

 It may be convenient to give the lattice a literal form by writing a for zero in the 

 compartments. 



Art. 10. Contrast the result obtained with that which arises from 



The lattices are similar to those above, with the additional condition that each row 

 is to contain but one letter a. Again, from 



arise lattices of t columns and s rows with the same condition as the zero lattices, 

 but with the additional conditions that the numbers of letters a in the successive 

 rows are to be p lt p.,, , . . p, respectively. This remark leads to a relationship 

 between the coefficients in the developments of (l*)(l")(l r ) . . . and (O x )(0")(0 > ') . . . 

 respectively. For let 



. . . p,) 



... p/) -f .... 



