MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS. Ill 



As examples, N(321) (-) +N(33) <-) = 3N(32) an) , 



derived from <?N { (32) ( 1 ) } (aw ; 



and N(24) (aM) +N(231) (aj) +N(213) (M2) +N(222) (af) 



= 6N(21) (m) +18N(21) (M) , 

 derived from 



Art. 71. The enumeration of the permutations, whose specifications contain a given 

 largest integer, will now be investigated. 



Let T .1 K 



AM) "mi rv - 



denote respectively 



in which 



(i.) the highest of the integers a, 6, c, ... w m or less ; 



(ii.) ,, .- or greater; 



(iii.) exactly; 



so that, when a + b + c+... = n, 



I w = 



J m .= K M 

 I. = J, = 



I m Im-l = "m~m + l = *! 



H (n) = N (n) = J. = K. = I.-T.-, ; 



. = J.-.+J. = -I.-.-I.-.+al. ; 



'118O 



)} = N(n-2, l') + N(n-l, l) + N(n-2, 2) + N(n), 



, n-l) + N(n), 



l, n- 

 and by addition 3 ^ w _ 2 )(l) 8 } = K._,+2K._, 



the law apparent here obtains so long as a number n-v appearing in 



N( ), 



