1898 - 99 .] Dr Muir on a Single Term of a Determinant. 447 
may be given as. 
4+1 +2+0+0, 
where 4 is the number of inverted-pairs in which the first element 
5 of the given permutation comes first, 1 the number in which the 
second element 2 of the given permutation comes first, and so on ; 
or, it may be given as 
3+1+2+1+0, 
where 3 is the number of inverted-pairs in which 1 comes second, 
1 the number in which 2 comes second, 2 the number in which 3 
comes second, and so on. 
The former of these two ways is that generally followed ; and, 
clearly, when this is done, the last item is necessarily 0, the second 
item from the end 1 or 0, the third item from the end 2 or 1 or 0, 
and so on. 
A modification of the former mode is got by arranging the items 
differently, viz., 
4 + 2 + 0+ 1 + 0, 
where 4 is the number of inverted-pairs in which the highest 
element 5 comes first, 2 the number in which the next highest 
element 4 comes first, and so on. If the inverted-pairs be written 
in rows according to this arrangement, thus 
51 52 53 54 
41 . 43 
21 . 
the columns so formed will give the items obtained by the second 
mode, viz., 
3+1+2+1+0. 
(9) If with a view to finding all the possible forms of 8 we 
try to fill n places subject to the conditions that the last place is 
to be filled with 0, the second from the end by 1 or 0, the third 
from the end by 2 or 1 or 0, and so on, we see that the filling of 
the places can be done in 1.2.3 n different ways. And, as 
this is exactly the number of different permutations of n things, 
it follows that the determination of the permutation from the 
detailed specification of the number of its inverted-pairs must be 
always possible. 
