98 MAJOR P. A. MACMAHON: MEMOIR ON THE 



It must be remembered, in assigning these successions to the space A, that only 

 ..in- dividing line is to be in the whole permutation, and that the latter must possess 

 the fundamental property which is the attribute of all such. 



Writing the above succession 



,",".. .a".- !., i . . ._!,, a B . 



we have to determine all of the successions ranging from 



.+! ..,-!,, to a u "- I <;}...<:|a r "" 1 



that may be placed between a. l n a a n ...a.\. l and the dividing line in order to form an LJ 

 succession involving <x r | u . 



Since , u + l, ..., r is a succession of v u+1 numbers in ascending order we are 

 clearly concerned with the whole of the partitions at the points of a one-row lattice 

 of v u+ 1 nodes which are such that the part magnitude lies between n 2 and zero. 



If i/ii/a---i/i>-i.+i be one such partition 



Denote by L, tC1 , that portion of Lj which is associated with the conjunction a c | a u ; 

 then 



leading to 



Put herein v = u+j, then, for a constant value of j, 



L, ( oo, m, nl = {a^ +1 + ay + " +1 + . +x --("-Do+i)x (n-1) (n) ... (n+j 1) 



(!)()... 0+1) 



L, ( oo, m, n) = 2 Z x *+(--" (n-1) (n) ... (n+v-u-1) 



consequently, giving j all values from 1 to m-I 



+g ,(nm-2n_) (n-l)(n)(n 

 ~ 



