1898 - 99 .] Dr Muir on a Single Term of a Determinant. 451 
pairs is 1 more than for the corresponding one of the third set of 
six permutations, 2 more than for the corresponding one of the 
second set of six permutations, and 3 more than for the corre- 
sponding one of the first set, which first set consists simply of the 
permutations of 12 3. Further, for either of the last two per- 
mutations of 12 3, the number of inverted-pairs is 1 more than 
for the corresponding one of the second set of two permutations, 
and 2 more than for the corresponding one of the first set, which 
first set consists simply of the permutations of 1 2. 
It follows, therefore, that from the numbers of inverted-pairs 
for the case of the permutations of 1 2, viz., 
0 , 1 , 
we can write the numbers of inverted-pairs for the case of the 
permutations of 1 2 3, viz., 
0 , 1 , 
1 , 2 , 
2,3; 
and that from these again we can write the numbers of inverted- 
pairs for the case of the permutations of 1 2 3 4, viz., 
0,1, 
1,2, 
2, 3, 
3, 4, 
1, 2, 
2, 3, 
3, 4, 
4, 5, 
2,3, 
3, 4, 
4, 5, 
5, 6. 
Consequently, if we denote by 
V M 
the number of permutations of 1 2 3 . . 
. n which have o inverted- 
pairs, we have 
V,»=l, 
v 3 ,„=i, 
V 4 , 0 =l, 
Vu = l, 
V,i = 2, 
Wx=3, 
V„=2, 
V 4 , 2 =5, 
v„=i, 
V 4 . s =6, 
V m =6, 
V 4 , 5 =3, 
V 4 , 6 =l. 
(14) Since in the case of the standard permutation 1 2 3 ... n 
