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  "body": "\n## Statement\n\nLet $p(n)$ be the [[partition function|001CX45SWQ]].\nThen, as formal power series in $\\mathbb{Z}[\\![x]\\!]$,\n$$\\sum_{n \\ge 0} p(n)\\, x^n = \\prod_{k \\ge 1} \\frac{1}{1 - x^k}.$$\n\n## Remarks\n\nThe infinite product makes sense, because each coefficient of the product depends only on finitely many factors.\nProof idea: the factor $\\frac{1}{1 - x^k} = 1 + x^k + x^{2k} + \\dots$ chooses how many parts equal $k$.\nRestricting the factors gives other results. For example, $\\prod_{k \\ge 1} (1 + x^k)$ counts partitions into distinct parts. This gives [[Euler's partition theorem|0XFZ1BMF5Q]].\n\n## Explanations\n\n### L1 human\n\nEach factor of the product decides how many times a part $k$ appears in the partition.\nMultiplying the factors combines these choices, and the coefficient of $x^n$ counts the choices with total $n$.\n",
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