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  "body": "\n## Definition\n\nLet $n \\geq 2$ and let $\\sigma \\in S_n$.\nWrite $\\sigma = \\tau_1 \\tau_2 \\cdots \\tau_m$ as a product of [[transpositions|0FWVRHBRWM]].\nThe sign of $\\sigma$ is $\\operatorname{sgn}(\\sigma) = (-1)^m$.\n\n## Remarks\n\nThe parity of $m$ does not depend on the decomposition, so the sign is well defined.\nEquivalently, $\\operatorname{sgn}(\\sigma) = (-1)^{N(\\sigma)}$, where $N(\\sigma)$ is the number of pairs $i \u003c j$ with $\\sigma(i) \u003e \\sigma(j)$.\nThe sign of a permutation is the determinant of its permutation matrix.\n\n## Explanations\n\n### L1 human\n\nEach permutation is a product of swaps, and the number of swaps is always even or always odd.\nThe sign records this: $+1$ for an even number and $-1$ for an odd number.\nA single swap has sign $-1$.\n",
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