460 Proceedings of Royal Society of Edinburgh. [sess. 
of combinatorials, all the values of V n ^ on the upper side of this 
diagonal will be expressible in the same way. Now such an 
aggregate is 
p — C 1 I p , p _ 
»+5-2,5 w+<$-3,<$-2 ^ ^+5 - 6,(5 - 5 n+<$-8 )( 5-7 
which when 8 = n becomes 
p _ p t p 1 n 
2 n - 2 ,n 2 n - 3,n - 2 " r 2n - 6,w - 5 ^2 n -8,n-7 
or 
C + C . p 
2 n - 3,?i " r 2n, - 6,n - 1 ^2 n - 8 ,n - 1 ' 
as it should be. Hence 
p _ p ,p I p _ p 
n+8-2,rc-2 n+S-3,re-l _t " ?i+S-6,n-l n+8-8,n-l ra+S-13,ra-l W ** 
is the correct expression of V n so long as 8 }> n. 
For example — 
"^ 11.8 — ^ 17,9 ^ 16.6 + ^ 13.3 + ^ 11 . 1 » 
= 24310-8008 + 286 + 11, 
= 24607-8008, 
= 16599. 
(23) The generating function of given in § 20 is already 
well known as the generating function of the numbers of com- 
binations of 1+2 + 3+ ... + (»— 1 ) things of which one is 
unique, two are alike but different from the first, three are alike 
but different from both the first and second sets, and so on. We 
have therefore the following curious proposition : — 
The number of permutations of 12 3 ... n which have 3 
inverted-pairs is equal to the number of 8-combinatiohs of Jn 
(n — 1) letters f one of ichich is a, tivo of which are b’s, three of which 
are c’s, and so forth. 
If we denote the number of such combinations by 
'M 
p 
^l+2+. ..+(«- 1),S . 
the proposition is, in symbols, 
y =c 
M 1+2+. . .+(n-l),a 
