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A023997
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Number of block permutations on an n-set.
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8
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1, 1, 3, 25, 339, 6721, 179643, 6166105, 262308819, 13471274401, 818288740923, 57836113793305, 4693153430067699, 432360767273547841, 44794795522199781243, 5176959027946049635225, 662704551840482536170579
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OFFSET
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0,3
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COMMENTS
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A block permutation of a set X is a bijection between two quotient sets of X (of necessarily equal rank).
Number of labeled partitions of (n,n) into pairs (i,j) where there are n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one black object and at least one white object. - Christian G. Bower, Jun 03 2005
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LINKS
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FORMULA
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a(0)=1, a(n) = Sum_{k=1..n} k! * S2(n,k)^2, S2(n,k) are the Stirling numbers of the second kind.
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EXAMPLE
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For n=3, there are the 3! ordinary permutations (of rank 3), 18 block permutations of rank 2 (2! for each pair of partitions of rank 2) and the single rank 1 one.
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MATHEMATICA
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Table[Sum[StirlingS2[n, k]^2k!, {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Jul 04 2011 *)
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PROG
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(Maxima) makelist(sum(stirling2(n, k)^2*k!, k, 0, n), n, 0, 24); /* Emanuele Munarini, Jul 04 2011 */
(PARI) a(n) = if (n==0, 1, sum(k=1, n, k!*stirling(n, k, 2)^2)); \\ Michel Marcus, Jun 18 2019
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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Des FitzGerald (D.FitzGerald(AT)utas.edu.au)
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EXTENSIONS
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STATUS
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approved
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