Determinants and Their Applications in Mathematical Physics

A.2 Permutations 309

wheremis the number of inversions required to transformJnintoIn,or

vice versa, by any method.σ=0ifJnis not a permutation ofIn.

Examples.

sgn

{

1234

2413

}

=− 1 ,

sgn

{

12345

35214

}

=1.

Permutations Associated with Pfaffians

Let the 2n-set{i 1 j 1 i 2 j 2 ···injn} 2 ndenote a permutation ofN 2 nsubject

to the restriction thatis<js,1≤s≤n. However, if one permutation

can be transformed into another by repeatedly interchanging two pairs

of parameters of the form{irjr}and{isjs}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.Putn= 2. There are three distinct permitted permutations ofN 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 permutationP 1 { 2314 }, for example, is excluded since it can be

transformed intoP{ 1423 }by interchanging the first and second pairs

of integers.P 1 is therefore not distinct fromPin this context.

b.Putn= 3. There are 15 distinct permitted permutations ofN 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 permutationsP 1 { 143625 }andP 2 { 361425 }, for example,

are excluded since they can be transformed intoP{ 142536 }by

interchanging appropriate pairs of integers.P 1 andP 2 are therefore not

distinct fromPin this context.

Lemma.

sgn

{

12 3 4... m

imr 3 r 4 ... rm

}

m