I0g MAJOR P. A. MACMAHON: MEMOIR ON THE 



For the whole of the permutations derived as above from the lattice which are, iu 

 fact, the whole of the permutations of ,W, we derive a permutation fund 



PF(oo;321), 



such that the generating function sought is 



PF(oo;32l) 

 (1)...(6) 



and this we know otherwise to have the value 



1 





m (2) (3) (4) (5) (6) 

 - 

 and, in general, 



an expression which is to be compared with the number which enumerates the 

 permutations of the letters in afaf ... /-. The former becomes equal to the latter 

 when x = 1. 



When the part magnitude is limited by the number I, the enumerating generating 



function of the partition is 



(l + l) ... (l + Pl ) . (l+l) ... (l + p 2 ) ...... (l + l) ... (l + Pn) 



but, from previous work, if PF, ( <x ; p^ . . . >) is the sub-permutation function 

 derived from the permutations possessing s dividing lines, this generating function 

 is also 



where v = Sp-p, (see ' Phil. Trans. Eoy. Soc.,' A, vol. 207, p. 119). 



Equating the two expressions for the generating function, and giving I the values 

 0, 1, 2, ... in succession, we find the relations 



1 = PF 0> I 



.- (Pn + D _ 



(1)" (1) 



(1) 



&c. =&c., 

 from which the general expression for PF, is readily obtainable. 



