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  "body": "\n## Statement\n\nLet $\\lambda$ be a [[partition|0JV30XN73Z]] of $n$ with [[Young diagram|096AHMB3P4]] $D$.\nFor a cell $c$ of $D$, let $a(c)$ be the number of cells of $D$ to the right of $c$ in its row.\nLet $l(c)$ be the number of cells of $D$ below $c$ in its column.\nThe hook length of $c$ is $h(c) = a(c) + l(c) + 1$.\nLet $f^\\lambda$ be the number of standard [[Young tableaux|Q6TH9PH2R7]] of shape $\\lambda$.\nThen\n$$f^\\lambda = \\frac{n!}{\\prod_{c \\in D} h(c)}.$$\n\n## Remarks\n\nThe formula is due to Frame, Robinson and Thrall.\nLet $F(\\lambda)$ denote the right-hand side of the formula.\nA short proof by Greene, Nijenhuis and Wilf uses a random \"hook walk\" inside the diagram.\nThey show that the walk ends at each corner $c$ with probability $F(\\lambda \\setminus c) / F(\\lambda)$, so these ratios add up to $1$.\nThe largest entry $n$ of a standard tableau sits in a corner.\nSo $f^\\lambda$ is the sum of $f^{\\lambda \\setminus c}$ over the corners $c$, and $F$ satisfies the same recursion; induction on $n$ finishes the proof.\n\nExample: for $\\lambda = (2, 2)$ the hook lengths are $3, 2$ in the first row and $2, 1$ in the second row.\nThe formula gives $4! / (3 \\cdot 2 \\cdot 2 \\cdot 1) = 2$, and indeed the standard tableaux are $\\begin{smallmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{smallmatrix}$ and $\\begin{smallmatrix} 1 \u0026 3 \\\\ 2 \u0026 4 \\end{smallmatrix}$.\nFor $\\lambda = (n)$ the hook lengths are $n, n-1, \\dots, 1$ and $f^\\lambda = 1$.\n\n## Explanations\n\n### L1 human\n\nThe formula counts the ways to fill a Young diagram with $1, \\dots, n$ so that rows and columns increase.\nEach box has a hook: the box itself, the boxes to its right and the boxes below it.\nDivide $n!$ by the product of all hook sizes and the answer is the count.\nIt is surprising that such a simple product gives an exact count.\n\n### L2 human\n\nThe number $f^\\lambda$ is the dimension of the irreducible representation of the symmetric group $S_n$ indexed by $\\lambda$.\nSo the hook length formula computes these dimensions directly.\nTogether with the [[Robinson–Schensted correspondence|1SCF8X47G1]], which gives $\\sum_{\\lambda} (f^\\lambda)^2 = n!$, it is a basic tool of the representation theory of $S_n$.\nThere are versions for shifted shapes and for trees, and a $q$-analogue that counts tableaux by a statistic.\n",
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