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  "body": "\n## Statement\n\nLet $n$ be a natural number.\nThen the number of [[partitions|0404R9P9J3]] of $n$ into odd parts equals the number of partitions of $n$ into distinct parts.\n\n## Remarks\n\nFor $n = 5$, the partitions into odd parts are $5$, $3 + 1 + 1$ and $1 + 1 + 1 + 1 + 1$. The partitions into distinct parts are $5$, $4 + 1$ and $3 + 2$.\nA proof with generating functions: $\\prod_{k \\ge 1} (1 + x^k) = \\prod_{k \\ge 1} \\frac{1 - x^{2k}}{1 - x^k} = \\prod_{k \\text{ odd}} \\frac{1}{1 - x^k}$.\nGlaisher gave a bijection. It merges equal parts in pairs, again and again, until all parts are distinct.\n\n## Explanations\n\n### L1 human\n\nCount the ways to write $n$ with only odd summands, and the ways to write it with summands that are all different.\nThe two counts are always equal, although the partitions themselves are not the same.\n",
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  "name": "Euler's partition theorem",
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