446 Proceedings of Boycd Society of Edinburgh. [sess. 
Year. 
Author. 
Things counted. 
Symbol for hTo. 
1750 
Cramer, 
inverted-pairs (derangements) 
8 
1800 
Rothe, . 
interchanges (V ertauschu ngen ) 
V 
1812 
Cauchy, 
circular substitutions 
K 
1831 
Drinkwater, . 
moves 
U 
1895 
Jenkins, 
even circular substitutions 
K e 
In the case of the third rule we have to remember that the sign- 
factor is not (-) K but ( - ) n ~ K . 
Our object is now to give some investigations regarding the five 
entities specified in the third column of this table. The “ Index 
of Contents ” at the end will show the main features of the work. 
INVERTED-PAIRS. 
(7) As regards “inverted-pairs,” it is manifest at the outset 
that the fewest possible number of them in any case is 0, which 
happens when the n elements are in their natural order ; and that 
the greatest possible number is (n — 1) + (n - 2) + . . . + 2 + I, i.e ., 
\n{yi - 1), which occurs when the natural order is reversed. 
Consequently, if T n>r be used to stand for the sum of the terms 
which have r inverted-pairs, the final expansion of the deter- 
minant is 
T W(0 - T n>1 + T n>2 -.... + ( 
Also, if Y^ be used to stand for the number of terms which have 
8 inverted-pairs, we have 
1-2.3 n = V nf0 + Y W|1 + Y w>2 + . • .+Y n i n (n-i)) 
or, more definitely 
J(w!) = Y w>0 + V nj2 + V n>4 + .... 
and J(«!) = Y w> i + Y % g + Y n>5 + .... 
(8) The full details of the number of inverted-pairs of a per- 
mutation — that is to say, the items which go to constitute 8 — 
may be specified in different orderly ways. Thus, the number of 
inverted-pairs of the permutation 
i-l)’ 
5 2 4 1 3 
