The pairs in inversion are the pairs of the form {I, K} with K ranging between i+1 and J and those of the form {K, J} with K ranging between i+1 and j-1 . On the whole, there is an odd number of inversions, and the disparity of the permutation results from this.

A formula for the signature

One notes $\{\backslash \; mathcal\; P\}$ the whole of the Paire S of elements ranging between 1 and N (there is N (n-1) of it /2 ). A permutation σ has as a signature

\ prod \ limits_ {\ {I, J \} \ in {\ mathcal P}} \ frac {\ sigma (I) - \ sigma (J)}{i-j}

Démonstration

Let us call P this product. To examine all the couples (I, J) with i amounts examining all the pairs {I, J} . For each one of them, the term which is in the product has a negative sign if the pair is in inversion, positive if not. This shows that the sign of P is well that of the signature. Lastly, by bijectivity of σ, the terms σ (I) - σ (J) of the numerator is, except for the sign, the same ones as the i-j of the denominator. This shows that the absolute value of P is worth 1 and makes it possible to conclude.

This formula has a certain algebraic interest but in practice does not allow an effective calculation of the signature. Indeed compared to the simple counting of the inversions is added the multiplication and division by a certain number of entireties.

Signature of a product

The permutations check a rule of the signs for the product: the product of two even permutations is even, of two odd permutations is even, the product of an even permutation and of odd is odd. By using the signature, that is summarized by the formula

$\backslash \; varepsilon\; (\backslash \; sigma\; \backslash \; circ\; \backslash \; tau)\; =\; \backslash \; varepsilon\; (\backslash \; sigma).\; \backslash \; varepsilon\; (\backslash \; tau)$

Demonstration

$\backslash \; varepsilon\; (\backslash \; sigma\; \backslash \; circ\; \backslash \; tau)\; =\; \backslash \; prod\; \backslash \; limits\_\; \{\backslash \; \{I,\; J\; \backslash \}\; \backslash \; in\; \{\backslash \; mathcal\; P\}\}\; \backslash \; frac\; \{\backslash \; sigma\; (\backslash \; tau\; (I))-\; \backslash \; sigma\; (\backslash \; tau\; (J))\}\{\backslash \; tau\; (I)\; -\; \backslash \; tau\; (J)\}$

\ prod \ limits_ {\ {I, J \} \ in {\ mathcal P}} \ frac {\ tau (I) - \ tau (J)}{i-j}

In the second term one recognizes a signature directly. For the first, one needs réindexer as a preliminary by posing {i', I} = {τ (I), τ (J)} , one then recognizes also a signature there.

In algebraic terms: the signature is a morphism of group S of the symmetrical Groupe $(\backslash \; mathfrak\; S\_n,\; \backslash \; circ)$ in $\backslash \; left\; (\backslash \; \{-\; 1,1\; \backslash \},\; \backslash \; times\; \backslash \; right)$. The whole of the even permutations forms the alternate Groupe, core of this morphism. Finally the permutation reverses σ with the same signature as σ.

Calculation of a signature

In corollary of the preceding results,

• a permutation is even if and only if it can be expressed like the product of an even number of transpositions;
• a permutation is odd if and only if it can be expressed like the product of an odd number of transpositions

and these two cases are excluded mutually.

The calculation of the signature by the decomposition in product of transpositions is much more effective than the application of the initial definition; indeed for a permutation of $\{\backslash \; mathfrak\; S\}\; \_n$ this decomposition requires of the more n-1 operations, against N (n-1) /2 for the definition.

; Examples

the identity is an even permutation;

a transposition is an odd permutation;

a circular Shift is even if the number of elements is odd; it is odd if the number of elements is even.

See too

• symmetrical Permutation

• Group

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