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  "body": "\n## Statement\n\nLet $n$ and $k$ be natural numbers, and let $c(n, k)$ be the unsigned [[Stirling numbers of the first kind|0Z22VQH17V]].\nThen\n$$c(n + 1, k + 1) = n\\, c(n, k + 1) + c(n, k).$$\n\n## Remarks\n\nProof idea: look at the element $n + 1$ in a permutation of $n + 1$ elements. Either it is a fixed point, and the rest has $k$ cycles. Or it sits right after one of the $n$ other elements in its cycle.\nExample: $c(4, 2) = 3\\, c(3, 2) + c(3, 1) = 3 \\cdot 3 + 2 = 11$.\n\n## Explanations\n\n### L1 human\n\nInsert a new element into a permutation written as cycles.\nIt forms a new cycle of length $1$, or it goes into an existing cycle right after one of the $n$ old elements.\n",
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  "name": "Recurrence for Stirling numbers of the first kind",
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