450 Proceedings of Royal Society of Edinburgh. [sess. 
ing value of 8 in detail, as has been done in § 10 for the case of 
n — 4. Then it will he seen that the first permutation whose 
detailed value of 8 begins with the item oq is preceded by cq sets 
of permutations, each set containing (n — 1)! permutations, so 
that there must precede the permutation in question cq.(w-l)! 
permutations at least. Following up the matter, however, we 
shall find for the same reason that the existence of the item a 2 
implies that a 2 .(n - 2)! more permutations precede the said per- 
mutation, and similarly in the case of a 3 , a 4 , . . . Consequently 
the number of this permutation is 1 more than 
a.(n — 1)! + a 2 .{n — 2)! + . . . + a n _ 1 .l! + a n 
as was to be proved. 
For example, the number of inverted-pairs being 
2+3+0+ 1 +0 
as in § 9, we multiply 2 by 4!, 3 by 3!, and 1 by 1!, take the sum 
of the products, and thus find the permutation to be the 67th. 
From this it is evident that if, on the other hand, the ordinal 
number of the permutation be known, we have only to take the 
integer next less than it, divide this integer by 2 and note the 
remainder, then divide the integral part of the quotient by 3 and 
note the remainder, then divide the integral part of the new 
quotient by 4 and note the remainder, and so on. This being 
done, the remainders are, when reversed in order, the items which 
constitute 8 for the permutation in question. For example, if it 
were the 54th permutation, the process would be 
2 
3 
4 
5 
53 
26, 1 
_ 8 , 2 
_ 1 ° 
0 , 2 
and 2 + 0 + 2 + 1 + 0 would be the required number of inverted- 
pairs in detailed form. 
(13) Looking again to the table of § 10 we see that for any one 
of the last six permutations of 1 2 3 4 the number of inverted- 
