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A326243
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Number of capturing set partitions of {1..n}.
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21
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0, 0, 0, 0, 1, 11, 80, 503, 2993, 17609, 105017, 644528
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OFFSET
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0,6
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COMMENTS
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A set partition is capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. This is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.
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LINKS
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FORMULA
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EXAMPLE
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The a(5) = 11 capturing set partitions:
{{1,2,5},{3,4}}
{{1,3,4},{2,5}}
{{1,3,5},{2,4}}
{{1,4},{2,3,5}}
{{1,4,5},{2,3}}
{{1,5},{2,3,4}}
{{1},{2,5},{3,4}}
{{1,4},{2,3},{5}}
{{1,5},{2},{3,4}}
{{1,5},{2,3},{4}}
{{1,5},{2,4},{3}}
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z&&y>t||x>z&&y<t];
Table[Length[Select[sps[Range[n]], capXQ[#]&]], {n, 0, 8}]
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CROSSREFS
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Non-capturing set partitions are A326254.
Crossing and nesting set partitions are (both) A016098.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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