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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.

Inversion

Let

x |
P_{1}(x) |
P_{2}(x) |
P_{3}(x) |
P_{4}(x) |
P_{5}(x) |
P_{6}(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 |
(a_{11} a_{22} a_{33}) |
(—a_{11} a_{23} a_{31}) |
(—a_{11} a_{21} a_{32}) |
(a_{12} a_{21} a_{32}) |
(a_{13} a_{21} a_{31}) |
(—a_{13} a_{22} a_{31}) |

**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**p**defined from_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}**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**p**_{2}, then we notice from the above table that**2 < i p(2) > p(1)**and as such**p**_{2}has one inversion. Obviously,**p**_{1}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.**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 FunctionSubmit Your Query ???

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