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  "body": "\n## Definition\n\nLet $n$ be a natural number.\nThe Bell number $B_n$ is the number of partitions of an $n$-element set into nonempty blocks.\nEquivalently, it is the sum of [[Stirling numbers of the second kind|0ZN07RQ3JS]]:\n$$B_n = \\sum_{k=0}^{n} S(n, k).$$\n\n## Remarks\n\nThe first values, for $n = 0, 1, \\dots, 6$, are $1, 1, 2, 5, 15, 52, 203$.\n$B_0 = 1$, because the empty set has exactly one partition, the partition with no blocks.\nThe values satisfy a [[recurrence|0110GBHPPB]] with binomial coefficients.\n\n## Explanations\n\n### L1 human\n\nThe Bell number $B_n$ counts all ways to split $n$ labelled objects into groups, with any number of groups.\nFor $\\{1, 2, 3\\}$ there are $5$ ways: $123$, $1|23$, $2|13$, $3|12$ and $1|2|3$.\n",
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