sign of a permutation

The sign of a permutation is a single +1 or -1 that records whether the permutation is built from an even or odd number of swaps. Shuffle a deck by a sequence of two-card swaps: no matter which sequence of swaps reaches a given arrangement, the COUNT of swaps is always even or always odd for that arrangement. The sign captures exactly that fixed parity. It is the lone source of the plus and minus signs inside every determinant.

A transposition is the simplest permutation: it swaps two elements and leaves the rest fixed. Every permutation factors into transpositions, and although the factorization is far from unique, its parity (even or odd) is an invariant. Define sign(sigma) = +1 if sigma decomposes into an even number of transpositions, and -1 if odd. Equivalently, sign(sigma) = (-1)^(number of inversions), where an inversion is a pair i < j with sigma(i) > sigma(j).

Why it matters here: the alternating axiom says swapping two arguments flips the determinant's sign. Apply that one swap at a time to undo a permutation, and you accumulate exactly sign(sigma) — which is why that factor sits in front of each Leibniz term. The sign is also a group homomorphism, sign(sigma tau) = sign(sigma) sign(tau), and that multiplicativity is the engine behind det(AB) = det(A) det(B).

Caveat: parity is well defined, but the NUMBER of transpositions is not — only its even/odd class is fixed. Do not try to read off the sign by counting an arbitrary swap sequence's length; count inversions, or factor into disjoint cycles where a cycle of length L contributes sign (-1)^(L-1).