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A326243
Number of capturing set partitions of {1..n}.
20
0, 0, 0, 0, 1, 11, 80, 503, 2993, 17609, 105017, 644528, 4107600, 27313805, 189866541, 1379728831, 10470032837, 82833202559, 681977545967, 5832430910181, 51723181525978, 474866750479993, 4506706112772881, 44151975623559477, 445958774322599940, 4638590033810841345
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
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
FORMULA
a(n) = A000110(n) - A054391(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_, ___}, ___}/; x<z&&y>t||x>z&&y<t];
Table[Length[Select[sps[Range[n]], capXQ[#]&]], {n, 0, 8}]
CROSSREFS
Non-capturing set partitions are A054391.
Crossing and nesting set partitions are (both) A016098.
Sequence in context: A026897 A021024 A127021 * A091098 A091115 A024146
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Jun 19 2019
EXTENSIONS
a(12) and beyond from Christian Sievers, Aug 23 2024
STATUS
approved