Page 323 - Determinants and Their Applications in Mathematical Physics

308 Appendix

and let I n and J n denote arrangements or permutations of the same n
integers
- .
I n = i 1 i 2 i 3 ··· i n ,
- .
J n = j 1 j 2 j 3 ··· j n .
There are n! possible sets of the form I n or J n including N n . The num-
bers within the set are called elements. The operation which consists of
interchanging any two elements in a set is called an inversion. Assuming
that J n = I n , that is, j r = i r for at least two values of r, it is possible to
transform J n into I n by means of a sequence of inversions. For example, it
is possible to transform the set {35214} into the set N 5 in four steps,
that is, by means of four inversions, as follows:
35214
1 : 15234
2 : 12534
3 : 12354
4 : 12345
The choice of inversions is clearly not unique for the transformation can
also be accomplished as follows:
35214
1 : 34215
2 : 31245
3 : 21345
4 : 12345
No steps have been wasted in either method, that is, the methods are
efficient and several other efficient methods can be found. If steps are wasted
by, for example, removing an element from its final position at any stage
of the transformation, then the number of inversions required to complete
the transformation is increased.
However, it is known that if the number of inversions required to trans-
form J n into I n is odd by one method, then it is odd by all methods, and
J n is said to be an odd permutation of I n . Similarly, if the number of in-
versions required to transform J n into I n is even by one method, then it is
even by all methods, and J n is said to be an even permutation of I n .
The permutation symbol is an expression of the form

i 1 i 2 i 3 ···
= i n ,
I n
j 1 j 2 j 3 ··· j n
J n
which enables I n to be compared with J n .
The sign of the permutation symbol, denoted by σ, is defined as follows:

i 1 i 2 i 3 ···
σ = sgn I n = sgn i n =(−1) ,
m
j 1 j 2 j 3 ··· j n
J n