

A326246


Number of crossing, capturing set partitions of {1..n}.


9



0, 0, 0, 0, 0, 3, 37, 307, 2173, 14344, 92402, 596688
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OFFSET

0,6


COMMENTS

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.


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.


EXAMPLE

The a(5) = 3 set partitions:
{{1,3,4},{2,5}}
{{1,3,5},{2,4}}
{{1,4},{2,3,5}}


MATHEMATICA

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<tz<x<t<y];
capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<t<yz<x<y<t];
Table[Length[Select[sps[Range[n]], capXQ[#]&&croXQ[#]&]], {n, 0, 5}]


CROSSREFS

MMnumbers of crossing, capturing multiset partitions are A326259.
Crossing set partitions are A016098.
Capturing set partitions are A326243.
Crossing, nesting set partitions are A326248.
Crossing, noncapturing set partitions are A326245.
Noncrossing, capturing set partitions are A122880 (conjecture).
Cf. A000108, A000110, A058681, A099947, A324170, A326211, A326249, A326254, A326255.
Sequence in context: A232303 A196978 A197163 * A036942 A183960 A338717
Adjacent sequences: A326243 A326244 A326245 * A326247 A326248 A326249


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Jun 20 2019


STATUS

approved



