Page 324 - Determinants and Their Applications in Mathematical Physics

A.2 Permutations 309

where m is the number of inversions required to transform J n into I n ,or
vice versa, by any method. σ =0 if J n is not a permutation of I n .
Examples.

1234
sgn = −1,
2413

12345
sgn =1.
35214
Permutations Associated with Pfaffians

Let the 2n-set {i 1 j 1 i 2 j 2 ··· i n j n } 2n denote a permutation of N 2n subject
to the restriction that i s <j s ,1 ≤ s ≤ n. However, if one permutation
can be transformed into another by repeatedly interchanging two pairs
of parameters of the form {i r j r } and {i s j s } then the two permutations
are not considered to be distinct in this context. The number of distinct
permutations is (2n)!/(2 n!).
n
Examples.
a. Put n = 2. There are three distinct permitted permutations of N 4 ,
including the identity permutation, which, with their appropriate signs,
are as follows: Omitting the upper row of integers,
sgn{1234} =1, sgn{1324} = −1, sgn{1423} =1.
The permutation P 1 {2314}, for example, is excluded since it can be
transformed into P{1423} by interchanging the first and second pairs
of integers. P 1 is therefore not distinct from P in this context.
b. Put n = 3. There are 15 distinct permitted permutations of N 6 , includ-
ing the identity permutation, which, with their appropriate signs, are
as follows:
sgn{123456} =1, sgn{123546} = −1, sgn{123645} =1,
sgn{132456} = −1, sgn{132546} =1, sgn{132645} = −1,
sgn{142356} =1, sgn{142536} = −1, sgn{142635} =1,
sgn{152346} = −1, sgn{152436} =1, sgn{152634} = −1,
sgn{162345} =1, sgn{162435} = −1, sgn{162534} =1.
The permutations P 1 {143625} and P 2 {361425}, for example,
are excluded since they can be transformed into P{142536} by
interchanging appropriate pairs of integers. P 1 and P 2 are therefore not
distinct from P in this context.
Lemma.

1 2 3 4 ... m
sgn
i m r 3 r 4
... r m
m