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Home » Math Homework Help » Algebra Homework Help » Permutation Function
Permutation Function
In this section we shall give the definition and some properties of permutation function which are useful in the development of the theory of determinants.

x P1(x) P2(x) P3(x) P4(x) P5(x) P6(x)
1 1 1 2 2 3 3
2 2 3 1 3 1 2
3 3 2 3 1 2 1
No. of inversion 0 1 1 2 2 3
Value of δ(p) + + +  
terms (a11 a22 a33) (—a11 a23 a31) (—a11 a21 a32) (a12 a21 a32) (a13 a21 a31) (—a13 a22 a31)

Definition: A one-one function whose domain and the range is the same set, the set being finite, is called a permutation function. For example, let S = {1, 2, 3} be a finite set, then there are 3 ! = 6 permutation functions p1, p2, p3, p4, p5, p6 defined from S to S. Let us explain the permutation functions by means of the above table.

Inversion

Let p be a permutation function and i < j be a pair of elements in its domain such that p (i) > p (j), then p is said to have an inversion. For example if S = {1, 2} and the permutation function is p2, then we notice from the above table that 2 < i p(2) > p(1) and as such p2 has one inversion. Obviously, p1 has zero inversion. In other words, an inversion is said to take place if in a permutation 3, 1 and 2, we have the couples (1, 2) which gives one inversion.

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