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  "body": "\n## Statement\n\nLet $\\sigma$ be a [[permutation|05RWFKVTJ7]] of $\\set{1, \\ldots, k}$ for some $k \\ge 1$.\nFor $n \\ge 1$, let $\\operatorname{Av}_n(\\sigma)$ be the set of permutations of $\\set{1, \\ldots, n}$ that avoid $\\sigma$ as a [[pattern|1H8Z1WMG23]].\nThen there is a constant $c_\\sigma$ such that $\\abs{\\operatorname{Av}_n(\\sigma)} \\le c_\\sigma^{\\,n}$ for all $n \\ge 1$.\n\n## Remarks\n\nThe conjecture was proved by Adam Marcus and Gábor Tardos in 2004.\nThey proved the Füredi–Hajnal conjecture on $0$-$1$ matrices that avoid a permutation matrix, and Martin Klazar had shown earlier that it implies the Stanley–Wilf conjecture.\nThe bound means that the limit $L(\\sigma) = \\lim_{n \\to \\infty} \\abs{\\operatorname{Av}_n(\\sigma)}^{1/n}$ is finite; Arratia showed that this limit exists.\nThe number $L(\\sigma)$ is the Stanley–Wilf limit of $\\sigma$.\n\nExample: for $\\sigma = 123$, $\\abs{\\operatorname{Av}_n(123)}$ is the [[Catalan number|0YWZJEABB2]] $C_n \\le 4^n$, so $c_{123} = 4$ works and $L(123) = 4$.\n\nWithout a forbidden pattern the statement fails, since there are $n!$ permutations of $\\set{1, \\ldots, n}$, and $n!$ grows faster than $c^n$ for each constant $c$.\n\n## Explanations\n\n### L1 human\n\nForbidding a single pattern is a strong restriction on a long permutation.\nThe theorem makes this precise: the number of permutations that avoid a fixed pattern grows only exponentially in $n$.\nWithout restriction there are $n!$ permutations, which grow much faster.\n\n### L2 human\n\nFor patterns of length $3$ the counts are Catalan numbers, but for longer patterns exact formulas are rare.\nThe theorem says that every pattern class has finite exponential growth rate, the Stanley–Wilf limit.\nThe proof by Marcus and Tardos is short and goes through extremal problems for $0$-$1$ matrices.\n",
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