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 A326243 Number of capturing set partitions of {1..n}. 21
 0, 0, 0, 0, 1, 11, 80, 503, 2993, 17609, 105017, 644528 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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. LINKS Table of n, a(n) for n=0..11. Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012. FORMULA a(n) = A000110(n) - A326254(n). EXAMPLE 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}} MATHEMATICA 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_, ___}, ___}/; xt||x>z&&y

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Last modified June 4 12:12 EDT 2023. Contains 363128 sequences. (Running on oeis4.)