452 
Proceedings of Royal Society of Edinburgh. [sess. 
none of the ^n{n- 1) pairs of elements is inverted, and in the 
case of 7?, n- 1, . . ., 2, 1 all of them are, it follows that, if we 
transform 1 2 3 ... n so as to obtain a permutation with 8 
inverted-pairs, the same transformation made upon n,n-l, . . ., 
2, 1 will produce a permutation with 8 uninverted-pairs, and 
therefore with ^n(n- 1 ) - 8 inverted-pairs. This implies that for 
every permutation of 1 2 3 . . . n with 8 inverted-pairs, there 
must thus be a permutation of nn - 1 . . . 2 1 with \n(n— 1) 
- 8 inverted-pairs. But the permutations of 12 3... n are 
exactly the permutations of nn — 1 ... 2 1 ; hence 
V n ,j = Vn,in(?i-l)-8 , 
as is seen in the preceding § to he the case for V 2 ,s, V 3 ,s> ^ 4 ,s . 
(15) Having got in § 10 the numbers of inverted-pairs for the 
24 permutations of 1 2 3 4, viz., 0, 1, 1, 2, 2, 3, . . ., we can im- 
mediately write the numbers for the 120 permutations of 1 2 3 4 5. 
All that is necessary is to copy out in one column the 24 numbers 
referred to, and then make four adjacent columns with each com- 
ponent number of any of the four greater by 1 than the corre- 
sponding number in the immediately preceding column : thus — 
It follows, of course, from this mode of constructing the five 
columns, that if we wish to know how often any number, say 7, 
occurs in the complete set of columns, we have only got to ascer- 
tain from previous work how often 7 occurs in the first column, 
and then recall the fact that 7 occurs in the second column exactly 
