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  "body": "\n## Example\n\nLet $n \\ge 1$ and let $S_n$ be the [[symmetric group|0515A6D2Y1]] of [[permutations|05RWFKVTJ7]] of $\\set{1, \\ldots, n}$, written in one-line notation $w = w(1) w(2) \\cdots w(n)$.\nThe inversion set $\\mathrm{Inv}(w)$ is the set of pairs of values $(a, b)$ with $a \u003c b$ such that $b$ stands left of $a$ in $w$.\nThe *weak order* on $S_n$ is defined by $u \\le w$ when $\\mathrm{Inv}(u) \\subseteq \\mathrm{Inv}(w)$.\nIt is a partial order, and it is a [[lattice|WH4SFAACAS]].\nThe covering relations are the moves that swap two adjacent entries $a \u003c b$ standing as $\\ldots a b \\ldots$.\nSuch a move adds the single pair $(a, b)$ to the inversion set.\nThe least element is the identity $12 \\cdots n$, and the greatest element is the reversal $n \\cdots 21$.\n\nCheck for $n = 3$.\nThe inversion sets are $\\mathrm{Inv}(123) = \\emptyset$, $\\mathrm{Inv}(213) = \\set{(1,2)}$, $\\mathrm{Inv}(132) = \\set{(2,3)}$, $\\mathrm{Inv}(231) = \\set{(1,2), (1,3)}$, $\\mathrm{Inv}(312) = \\set{(1,3), (2,3)}$, and $\\mathrm{Inv}(321)$ has all three pairs.\nSo the weak order is the two chains $123 \u003c 213 \u003c 231 \u003c 321$ and $123 \u003c 132 \u003c 312 \u003c 321$.\nThe elements $213$ and $312$ are incomparable, since $(1,2) \\notin \\mathrm{Inv}(312)$.\n\n## Remarks\n\nThe weak order is graded by the number of inversions.\nIts Hasse diagram for $S_3$ is a hexagon.\nThe Bruhat order has more relations: there $213 \u003c 312$.\n\n## Explanations\n\n### L1 human\n\nStart with the numbers in order, $1 2 \\cdots n$, and swap neighbours that are in the right order, one swap at a time.\nEach swap creates one new out-of-order pair.\nA permutation is below another when the pairs it has out of order are among those of the other.\nFor three numbers you get a hexagon, from $123$ at the bottom to $321$ at the top.\n",
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