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A326245
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Number of crossing, non-capturing set partitions of {1..n}.
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6
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0, 0, 0, 0, 1, 7, 34, 141, 537, 1941, 6777, 23096, 77340
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OFFSET
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0,6
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COMMENTS
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A set partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y, and capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. Capturing 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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EXAMPLE
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The a(4) = 1 and a(5) = 7 set partitions:
{{1,3},{2,4}} {{1,2,4},{3,5}}
{{1,3},{2,4,5}}
{{1},{2,4},{3,5}}
{{1,3},{2,4},{5}}
{{1,3},{2,5},{4}}
{{1,4},{2},{3,5}}
{{1,4},{2,5},{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, ___}];
croXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<t<y||z<x<y<t];
Table[Length[Select[sps[Range[n]], !capXQ[#]&&croXQ[#]&]], {n, 0, 10}]
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CROSSREFS
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Crossing set partitions are A016098.
Non-capturing set partitions are A326254.
Crossing, capturing set partitions are A326246.
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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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