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  "body": "\n## Definition\n\nThe partition function is the function $p : \\mathbb{N} \\to \\mathbb{N}$ such that $p(n)$ is the number of [[partitions|0404R9P9J3]] of $n$.\n\n## Remarks\n\nThe values for $n = 0, 1, \\dots, 10$ are $1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42$.\nThe [[generating function for partitions|0578PA53VA]] is $\\sum_{n \\ge 0} p(n) x^n = \\prod_{k \\ge 1} \\frac{1}{1 - x^k}$.\nThe function grows faster than any polynomial and slower than any exponential $c^n$ with $c \u003e 1$.\n\n## Explanations\n\n### L1 human\n\n$p(n)$ counts the ways to write $n$ as a sum of positive integers in any order.\nFor example, $p(5) = 7$: $5$, $4+1$, $3+2$, $3+1+1$, $2+2+1$, $2+1+1+1$ and $1+1+1+1+1$.\n",
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