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  "body": "\n## Definition\n\nLet $\\lambda = (\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge \\lambda_\\ell \u003e 0)$ be a [[partition|0JV30XN73Z]] of $n$.\nLet $D$ be its [[Young diagram|096AHMB3P4]], the set of cells $(i, j)$ with $1 \\le i \\le \\ell$ and $1 \\le j \\le \\lambda_i$, where $i$ is the row and $j$ the column.\n\n1. A *Young tableau* of shape $\\lambda$ is a map $T \\colon D \\to \\set{1, 2, 3, \\dots}$, called a filling of $D$.\n2. $T$ is *semistandard* if $T(i, j) \\le T(i, j+1)$ and $T(i, j) \u003c T(i+1, j)$ whenever both cells lie in $D$. So entries weakly increase along rows and strictly increase down columns.\n3. $T$ is *standard* if it is semistandard and $T$ is a bijection from $D$ onto $\\set{1, \\dots, n}$.\n\n## Remarks\n\nExample: for $\\lambda = (2, 2)$, the filling with rows $1\\,2$ and $3\\,4$ is standard.\nEach row increases ($1 \u003c 2$, $3 \u003c 4$), each column increases ($1 \u003c 3$, $2 \u003c 4$), and each of $1, 2, 3, 4$ appears once.\nThe filling with rows $1\\,1$ and $2\\,3$ is semistandard but not standard, because $1$ appears twice.\n\nNon-example: the filling of shape $(2, 1)$ with first row $2\\,1$ and second row $3$ is not semistandard, because the first row decreases.\n\nMany authors say \"tableau\" for a semistandard tableau.\nThe *content* of a semistandard tableau $T$ is the sequence $(\\mu_1, \\mu_2, \\dots)$, where $\\mu_k$ is the number of entries equal to $k$.\nThe number $f^\\lambda$ of standard tableaux of shape $\\lambda$ is given by the [[hook length formula|VCMS1CP91N]].\n",
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