464 Proceedings of Royal Society of Edinburgh. [sess. 
inverted-pair of the conjugate permutation. Similarly the un- 
inverted-pair 39 in the given permutation corresponds to the 
uninverted-pair 15 in the conjugate permutation. 
From this it follows that conjugate permutations have the same 
number of inverted-pairs, and therefore have the same sign — which 
is Rothe’s proposition. 
Moves. 
(28) The next subject which it is convenient to consider is that 
of “moves,” and at the outset it is important to note that whereas 
the number of inverted-pairs in any given permutation is definite, 
the number of moves which may he used to transform a given per- 
mutation into the standard permutation is in general indefinite, 
because of the possibility of proceeding in divers ways in making 
the moves. If it be agreed, however, to make the moves in 
orderly fashion, viz., so as to put the elements in order into their 
standard places, the number is quite definite, and it is this number 
which g is used to denote. Thus, for the permutation 
2 4 15 3 
we have /x = 4, because to attain the standard permutation it is 
necessary to put 1 into its standard place — which requires 2 moves, 
and then 3 into its standard place — which requires other 2. 
Of course, we might put the elements in reverse order into their 
standard places, but, as will be seen immediately, the total number 
of moves would not then be different. Thus, for the permutation 
2 4 1 5 3, we should have to put 5 into its standard place, then 
4, and then 2, the number of necessary moves thus being 1 + 2 + 1, 
?.e., 4 as before. 
(29) The fundamental proposition in regard to “ moves ” is the 
following : — 
The number of orderly moves necessary to transform any given 
permutation of the first n integers into the standard permutation is 
equal to the number of inverted-pairs in the former ; i.e., in symbols , 
/x = S . 
In the given permutation let a,/3,y, ... be the numbers of “in- 
