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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.
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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| 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
Services: - Permutation Function Homework | Permutation Function Homework Help | Permutation Function Homework Help Services | Live Permutation Function Homework Help | Permutation Function Homework Tutors | Online Permutation Function Homework Help | Permutation Function Tutors | Online Permutation Function Tutors | Permutation Function Homework Services | Permutation Function
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