Permutation Sign Calculator

About this calculator

The sign (or parity) of a permutation of 1..n tells you whether it can be built from an even or odd number of pairwise swaps (transpositions) of the identity order. This calculator counts inversions — pairs of positions where a larger number appears before a smaller one — and reports even inversions as sign +1, odd as sign −1.

For example, the permutation (2,1,3) has exactly one inversion (2 before 1), so it's odd with sign −1, while (2,3,1) has two inversions and is even with sign +1. This is exactly the quantity that appears in the Leibniz formula for a matrix determinant, where each term in the sum is weighted by the sign of the permutation defining that term.

Linear algebra and combinatorics students use it to check determinant expansions by hand, and it comes up in group theory when classifying elements of the symmetric group S_n as belonging to the alternating group (even permutations only).