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  "body": "\n## Definition\n\nLet $X$ be a set.\nA permutation of $X$ is a bijection $\\sigma : X \\to X$.\n\nThe permutations of $X$ form a [[group|G5S1M9]] under composition, with the identity map as identity element.\nThis group is the symmetric group on $X$, written $\\operatorname{Sym}(X)$.\nFor $X = \\{1, \\dots, n\\}$ it is written $S_n$.\n\n## Remarks\n\nFor a finite set $X = \\{1, \\dots, n\\}$, a permutation is often written in one-line notation as the sequence $\\sigma(1) \\sigma(2) \\cdots \\sigma(n)$. In this form it is an arrangement of the elements in a row.\nThe [[number of permutations|0CEBCS0E0E]] of an $n$-element set is $n!$.\nA [[derangement|0TH61JR04G]] is a permutation with no fixed point.\n\n## Explanations\n\n### L1 human\n\nA permutation is a rearrangement of the elements of a set.\nFor $\\{1, 2, 3\\}$, the map that sends $1 \\mapsto 2$, $2 \\mapsto 3$ and $3 \\mapsto 1$ is a permutation. In one-line notation it is $231$.\n",
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