Page 322 - Determinants and Their Applications in Mathematical Physics

A.2 Permutations 307

j S ij
1 2 3 4 5
i
1 1
2 1 1
3 1 3 1
4 1 7 6 1
5 1 15 25 10 1
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 matrices s n (x) and S n (x) are defined as follows:
1
 
 −x 1 
 2x 2 −3x 1 


s n (x)= s ij x i−j 3 2  ,

 −6x 11x −6x 1
= 
n 
24x −50x 35x −10x 1
 4 3 2 
...............................
n
1
 
 x 1 
 x 2 3x 1 


S n (x)= S ij x i−j =  3 2  .
 x 7x 6x 1
n 
x 15x 25x 10x 1
 4 3 2 
.........................
n
A.2 Permutations
Inversions, the Permutation Symbol
The first n positive integers 1, 2, 3,...,n, can be arranged in a linear se-
quence in n! ways. For example, the first three integers can be arranged in
3! = 6 ways, namely
123
132
213
231
312
321
Let N n denote the set of the first n integers arranged in ascending order of
magnitude,
- .
N n = 123 ··· n ,