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A326246
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Number of crossing, capturing set partitions of {1..n}.
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9
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0, 0, 0, 0, 0, 3, 37, 307, 2173, 14344, 92402, 596688
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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(5) = 3 set partitions:
{{1,3,4},{2,5}}
{{1,3,5},{2,4}}
{{1,4},{2,3,5}}
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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, 5}]
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CROSSREFS
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MM-numbers of crossing, capturing multiset partitions are A326259.
Crossing set partitions are A016098.
Capturing set partitions are A326243.
Crossing, nesting set partitions are A326248.
Crossing, non-capturing set partitions are A326245.
Non-crossing, capturing set partitions are A122880 (conjecture).
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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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