Determinants and Their Applications in Mathematical Physics

A.2 Permutations 307

Sij

j

i 12345

1 1

2 11

3 131

4 1761

5 11525101

Further values are given by Abramowitz and Stegun. Stirling numbers ap-

pear in Section 5.6.3 on distinct matrices with nondistinct determinants

and in Appendix A.6.

The matricessn(x) andSn(x) are defined as follows:

sn(x)=

[

sijx

i−j

]

n

=

      

1

−x 1

2 x

2

− 3 x 1

− 6 x

3

11 x

2

− 6 x 1

24 x

4

− 50 x

3

35 x

2

− 10 x 1

...............................

       n

,

Sn(x)=

[

Sijx

i−j

]

n

=

      

1

x 1

x

2

3 x 1

x

3

7 x

2

6 x 1

x

4

15 x

3

25 x

2

10 x 1

.........................

       n

.

A.2 Permutations

Inversions, the Permutation Symbol

The firstnpositive integers 1, 2 , 3 ,...,n, can be arranged in a linear se-

quence inn! ways. For example, the first three integers can be arranged in

3! = 6 ways, namely

123

132

213

231

312

321

LetNndenote the set of the firstnintegers arranged in ascending order of

magnitude,

Nn=

{

123 ···n

}

,