1898-99.] Dr Muir on a Single Term of a Determinant. 453 
as often as 6 occurs in the first, that 7 occurs in the third column 
exactly as often as 5 occurs in the first, and so on. Consequently, 
denoting as before by V 5J the number of permutations of 
1 2 3 4 5 in which there are 7 inverted-pairs, we have 
y 5>7 = y 4>7 + y 4 . 6 + y 4>5 + y 4>4 + y 4>3 , 
= 0+1+ 3+5 + G, 
= 15; 
or, using the theorem of § 14, 
v 5)7 =y 5i3 , 
=v 4>3 +y 4>2 +y 4)1 +y 4> 0, 
= 6+ 5+ 3+ 1, 
= 15. 
Making now one continuous column of the above 120 numbers 
by putting the second column under the first, the third under the 
new position of the second, and so on, we should obtain the first 
120 of the 720 numbers of inverted-pairs wanted for the case 
of the permutations of 1 2 3 4 5 6, the 600 others being got by 
forming five adjacent columns as before. It is clear, therefore, that 
generally we have 
Y’n J s = Y, l _i,fi + Y re -l,S-l + Yrc_i j s_2 +.... + V w -l,8_»+i, 
the number of terms on the right being n. 
(16) With this difference-equation, and the knowledge that 
y ?l)0 =l and Yi,s = 0 (except when 8 = 0), the accompanying table 
is easily constructed. 
[Table. 
