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  "body": "\n## Statement\n\nLet $m, n \\ge 1$.\nLet $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] with nonnegative integer entries.\nList the pairs $(i, j)$ in lexicographic order, each pair repeated $a_{ij}$ times.\nInsert the second coordinates $j$ in this order into the empty tableau by row insertion, and let $P(A)$ be the result.\nLet $Q(A)$ be the tableau that has entry $i$ in the cell created when the pair $(i, j)$ is inserted.\n\n1. $P(A)$ and $Q(A)$ are semistandard [[Young tableaux|Q6TH9PH2R7]] of the same shape.\n2. The entry $j$ appears in $P(A)$ exactly $\\sum_i a_{ij}$ times, and the entry $i$ appears in $Q(A)$ exactly $\\sum_j a_{ij}$ times.\n3. The map $A \\mapsto (P(A), Q(A))$ is a bijection onto the set of pairs $(P, Q)$ of semistandard tableaux of the same shape. Here $P$ has entries in $\\set{1, \\dots, n}$ and $Q$ has entries in $\\set{1, \\dots, m}$.\n4. $P(A^T) = Q(A)$ and $Q(A^T) = P(A)$.\n\n## Remarks\n\nThe insertion rule is the one of the [[Robinson–Schensted correspondence|1SCF8X47G1]]: an inserted entry $x$ bumps the leftmost entry of the row that is strictly larger than $x$.\nEqual first coordinates are listed with increasing second coordinates, so the recording tableau stays semistandard.\nFor a permutation matrix the construction gives back the Robinson–Schensted correspondence.\n\nExample: for $A = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ the pairs are $(1,1), (1,2), (2,2)$.\nInserting $1, 2, 2$ gives $P = \\begin{smallmatrix} 1 \u0026 2 \u0026 2 \\end{smallmatrix}$, and $Q = \\begin{smallmatrix} 1 \u0026 1 \u0026 2 \\end{smallmatrix}$.\nThe column sums $1, 2$ of $A$ match the content of $P$, and the row sums $2, 1$ match the content of $Q$.\n\n## Explanations\n\n### L1 human\n\nThe correspondence turns a table of counts into a pair of semistandard tableaux of the same shape, and back.\nThe column totals of the table become the numbers of $1$s, $2$s, and so on in one tableau.\nThe row totals become the same numbers for the other tableau.\n\n### L2 human\n\nSumming over all matrices with weight $\\prod_{i,j} (x_i y_j)^{a_{ij}}$ gives the Cauchy identity\n$$\\prod_{i=1}^{m} \\prod_{j=1}^{n} \\frac{1}{1 - x_i y_j} = \\sum_{\\lambda} s_\\lambda(x_1, \\dots, x_m) \\, s_\\lambda(y_1, \\dots, y_n),$$\nwhere $s_\\lambda$ is the [[Schur polynomial|43VFMJCDYJ]].\nPart 4 applied to symmetric matrices gives similar identities for $\\sum_\\lambda s_\\lambda$.\nThe correspondence is due to Knuth, and it is the main bijective tool in the theory of symmetric functions.\n",
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