1898-99.] Dr Muir on a Single Term of a Determinant. 463 
1234. 
2134,1324,1243. 
2314,3124:2143,1342:1423. 
4321. 
and this new column is seen to contain the 24 permutations of 
1 2 3 4 separated into seven sets, viz., those which have no 
inverted-pairs, those which have one, those which have two, and 
so forth. 
(27) A comparison of the number of inverted-pairs in one per- 
mutation with the number in the conjugate permutation* was 
originally made by Rothe in the paper above referred to. The 
following simple fundamental theorem, however, puts the matter 
in a fresh light : — 
If in any permutation of 1, 2, 3, . . . , n any one of the elements X 
be in the 1 th place and any other y he in the m th place , then accord- 
ing as Xy is or is not an inverted-pair there will he in the conjugate 
permutation an inverted-pair ml or an uninverted-pair lm. 
To say that Xy is an inverted-pair of the original permutation is 
the same as to say that X>y and precedes y, that is, that 
X > y and m > l. 
Now, in the conjugate permutation the element l will be in the 
X th place and the element m in the y th place : consequently, since 
X > y the element m will in that permutation precede the element 
l : and therefore, since m > Z, ml will be an inverted-pair. The 
reasoning is exactly similar when Xy is an uninverted-pair. 
Thus in the permutation 
367892154 
75 is an inverted-pair, and since the places of these two elements 
are respectively the 3rd and 8th, it follows that 83 must be an 
* “ Two permutations of the numbers 1, 2, 3, . . n are called conjugate 
when each number and the number of the place which it occupies in the one 
permutation are interchanged in the case of the other permutation.” See 
Muir, “ History of Determinants,” pp. 59, 60 ; Muir, “ On Self-Conjugate 
Permutations.” — Proc. Roy. Soc. Edin ., xvii. pp. 7-13. 
