94 MAJOR P. A MACMAHON: MEMOIR ON THE 



This leads to the new result 



= 

 and as a particular case, 



GF( oo; nn) = GF(oo, 2, ) = (1){(8 )(3)..! (n)}(n+l) ' 



a result already known. 



Art. 17. The determination of L(oo; ale) presents great difficulties, so that the 

 investigation proceeds in the path of least resistance. When the lattice is complete, 

 the Greek letter succession is conveniently taken to be 



It is clear that each permutation, that arises from the lattice, must terminate with 

 tt m ; hence this latter may be always deleted, and we find 



L(oo; n"' 1 , n-l) = L(oo; '") 



and the sub-lattice functions are also equal, but the inner lattice functions differ ; 

 thus it will be found that 



IL ( oo; nn) = 1 



but 



IL ( oo; n, n\) = 1+aj". 



The Sub-lattice Functions. 



Art. 18. It is necessary to inquire as to the highest order of sub-lattice function 

 that presents itself. For a lattice of TO rows and n columns I form the rectangular 

 scheme 



1 











where there are n columns. 



Reading this parallel to the arrow (inclined at 45 degrees), commencing with the 

 left-hand top corner, I obtain the permutation 







This is the permutation which involves the maximum number of dividing lines and 

 to the sub-lattice function of highest order; the permutation is unique, 

 mg a single power of *, which is the sub-lattiqe function in question. The 

 dividing lines may be counted, 



