1898-99.] Dr Muir on a Single Term of a Determinant. 449 
(11) A glance at the details under the heading “No. of 
Inverted-Pairs ” shows that the first column of items consists of 
6 ( i.e ., 3!) zeros, 6 ones, 6 twos, 6 threes; that the second column 
consists of 2 (i.e., 2!) zeros, 2 ones, 2 twos, followed repeatedly by 
the same; and that the third column consists of 1 (i.e., 1!) zero, 
1 one, followed repeatedly by the same. The explanation of this 
is that if in the case of the first, second, third, or fourth set of six 
permutations we strike off the first column of items of the 
inverted-pairs we obtain the reduced column 
0 + 0 + 0 
0+1+0 
1+0 + 0 
1 + 1 + 0 
2 + 0 + 0 
2 + 1 + 0 , 
which are the numbers of inverted-pairs for the permutations of 
234, or 134, or 124, or 123; and that if we treat this reduced 
column in the same way we obtain 
0 + 0 
1 + 0, 
which are the numbers of inverted-pairs for the permutations 
of 12. 
(12) It follows from this that each of the integers from 1 up 
to n\ has corresponding to it a special form of 8, that therefore 
when any integer is given we ought to be able to determine the 
corresponding value of 8 in full detail, and conversely, that when 
the details of 8 are given the number of the corresponding per- 
mutation should be obtainable. 
The theorem which effects the latter determination is — 
If for the N th permutation of n elements the number of inverted- 
pairs he 
<Xi + a 2 + a 3 + . . . . + a n 
then 
N — l = a 1 .(?i — 1)! + a 2 .(n — 2)! + . . . . + a^.j.l! + a n . 
Let all the n ! permutations be arranged in order in a column, 
and opposite each permutation in another column the correspond- 
