Parity of a permutation

If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation

The signature defines the alternating character of the symmetric group Sn.

Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non-bijective maps.

The sign of a permutation can be explicitly expressed as where N(σ) is the number of inversions in σ. Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as where m is the number of transpositions in the decomposition.

Following the method of the cycle notation article, this could be written, composing from right to left, as There are many other ways of writing σ as a composition of transpositions, for instance but it is impossible to write it as a product of an even number of transpositions.

[1] An even permutation can be obtained as the composition of an even number (and only an even number) of exchanges (called transpositions) of two elements, while an odd permutation can be obtained by (only) an odd number of transpositions.

The following rules follow directly from the corresponding rules about addition of integers:[1] From these it follows that Considering the symmetric group Sn of all permutations of the set {1, ..., n}, we can conclude that the map that assigns to every permutation its signature is a group homomorphism.

[2] Furthermore, we see that the even permutations form a subgroup of Sn.

[4] The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a coset of An (in Sn).

This follows from formulas like In practice, in order to determine whether a given permutation is even or odd, one writes the permutation as a product of disjoint cycles.

The permutation (1 2)(3 4) in A4 shows that the converse is not true in general.

This section presents proofs that the parity of a permutation σ can be defined in two equivalent ways: Let σ be a permutation on a ranked domain S. Every permutation can be produced by a sequence of transpositions (2-element exchanges).

In other words, the parity of the number of inversions of a permutation is switched when composed with an adjacent transposition.

An alternative proof uses the Vandermonde polynomial So for instance in the case n = 3, we have Now for a given permutation σ of the numbers {1, ..., n}, we define Since the polynomial

Suppose we want to swap the ith and the jth element.

Now if we count the inversions gained (or lost) by swapping the ith and the jth element, we can see that this swap changes the parity of the count of inversions, since we also add (or subtract) 1 to the number of inversions gained (or lost) for the pair (i,j).

From a decomposition into m disjoint cycles we can obtain a decomposition of σ into k1 + k2 + ... + km transpositions, where ki is the size of the ith cycle.

Suppose a transposition (a b) is applied after a permutation σ.

after t2 after ... after tr after the identity (whose N is zero) observe that N(σ) and r have the same parity.

Parity can be generalized to Coxeter groups: one defines a length function ℓ(v), which depends on a choice of generators (for the symmetric group, adjacent transpositions), and then the function v ↦ (−1)ℓ(v) gives a generalized sign map.