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A124104
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Sum of the Rand distance between all pairs of set partitions of {1, 2, ... n}.
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1
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0, 2, 36, 600, 11100, 235560, 5746524, 160252456, 5069446560, 180479494440, 7177165063750, 316636751823480, 15401586272510880, 821382267765103590, 47788292465454829260, 3019446671476746981600, 206339807951889894605488
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OFFSET
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1,2
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COMMENTS
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Given a partition of {1, ..., n}, look at a pair of elements. If the two elements are in the same block of the partition, they're called "co-clustered". The Rand distance between two partitions then counts the pairs that are co-clustered in exactly one of the two partitions. The Rand index is found by dividing the Rand distance by (n choose 2).
Example: The distance from 12 3 4 to 1 234 is 4 because of the four pairs 12 (in the first partition but not the second) and 23, 24, 34 (in the second partition but not the first). The maximal distance of 6 is attained by 1 2 3 4 and 1234. The Rand distance has some nice properties, satisfies the triangle inequality and there are linear-time algorithms for computing it. (End)
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LINKS
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FORMULA
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a(n) = 2 * binomial(n,2) * Bell(n-1) * (Bell(n) - Bell(n-1)).
a(n) ~ n*LambertW(n)*Bell(n)^2 * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021
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EXAMPLE
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E.g. a(2) = 2 = 1 + 1 + 0 + 0 because the distance from 1,2 to 12 is 1 (and vice versa) and the distance from 1,2 to 1,2 or 12 to 12 is 0.
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MATHEMATICA
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Table[2 Binomial[n, 2]*BellB[n - 1] (BellB[n] - BellB[n - 1]), {n, 17}] (* Michael De Vlieger, Apr 16 2015 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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