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  "body": "\n## Summary\n\nA Young tableau is a [[Young diagram|096AHMB3P4]] whose boxes are filled with positive integers, with the entries increasing along rows and down columns.\nIn a standard tableau the entries are $1, \\dots, n$, each used once; in a semistandard tableau repetitions are allowed along rows.\nTableaux index bases of the irreducible representations of the symmetric group and of the general linear group.\nThey are the combinatorial heart of [[Schur polynomials|43VFMJCDYJ]], the [[hook length formula|VCMS1CP91N]] and the [[Robinson–Schensted correspondence|1SCF8X47G1]].\n\n## Explanations\n\n### L1 human\n\nDraw the rows of boxes of a [[partition|0JV30XN73Z]], longest row on top.\nNow write a number in each box so that numbers grow when you move right and when you move down.\nFor the shape with rows of lengths $2$ and $1$, the fillings of $1, 2, 3$ that work are $\\begin{smallmatrix} 1 \u0026 2 \\\\ 3 \u0026 \\end{smallmatrix}$ and $\\begin{smallmatrix} 1 \u0026 3 \\\\ 2 \u0026 \\end{smallmatrix}$.\nCounting such fillings answers questions in algebra, probability and the study of permutations.\n",
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