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  "body": "\n## Statement\n\nLet $n$ be a positive integer.\nThen the number of [[compositions|0J8ANSYKAC]] of $n$ is $2^{n-1}$.\n\n## Remarks\n\nProof: write $n$ as a row of $n$ ones. A composition is a choice of where to cut among the $n - 1$ gaps between them.\nThe number of compositions of $n$ into exactly $k$ parts is $\\binom{n-1}{k-1}$.\nFor $n = 0$ there is one composition, the empty one.\n\n## Explanations\n\n### L1 human\n\nPut $n$ dots in a row. Between two neighbours you either draw a bar or you do not.\nThere are $n - 1$ gaps with $2$ choices each, so there are $2^{n-1}$ compositions.\n",
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