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  "body": "\n## Statement\n\nLet $t$ be a positive integer, and let $R = (r_1 \\ge r_2 \\ge \\dots)$ and $C = (c_1 \\ge c_2 \\ge \\dots)$ be two [[partitions|0404R9P9J3]] of $t$.\nA $0$-$1$ [[matrix|0N1MY1T0W9]] has row sums $R$ and column sums $C$ when it has one row for each part of $R$ and one column for each part of $C$, its entries are all $0$ or $1$, row $i$ adds up to $r_i$, and column $j$ adds up to $c_j$.\n\nLet $R^d$ be the conjugate partition of $R$. Its $j$-th part is the number of parts of $R$ that are at least $j$.\n\nThen a $0$-$1$ matrix with row sums $R$ and column sums $C$ exists if and only if $R^d$ dominates $C$ in the dominance order on partitions of $t$.\nThat is, for each $k \\ge 1$, the sum of the $k$ largest parts of $R^d$ is at least the sum of the $k$ largest parts of $C$, where missing parts count as $0$.\n\n## Remarks\n\nThe dominance condition is the [[majorization|5K7VT2VKZT]] of $C$ by $R^d$, applied to partitions of the same number.\nExample: $R = (2, 2)$ and $C = (2, 2)$ give $R^d = (2, 2)$, which dominates $C$, and the matrix with all entries $1$ works.\nExample: $R = (2, 2)$ and $C = (4)$ give $R^d = (2, 2)$, which does not dominate $(4)$, since $2 \u003c 4$. No such matrix exists, as a column with $2$ rows cannot sum to $4$.\n",
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    "kind": "statement",
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    "name": "Gale–Ryser theorem",
    "role": "theorem",
    "sources": [
      {
        "locator": "source/s_kitchensink_zero-one-matrices.ptx:60",
        "url": "https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_kitchensink_zero-one-matrices.ptx#L60"
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  "markdown": "https://math2.tamnd.com/node/5DFX9M41GY.md",
  "msc": [
    "05B20",
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  "name": "Gale–Ryser theorem",
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  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5DF/5DFX9M41GY.md",
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