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A326245 Number of crossing, non-capturing set partitions of {1..n}. 6
0, 0, 0, 0, 1, 7, 34, 141, 537, 1941, 6777, 23096, 77340 (list; graph; refs; listen; history; text; internal format)
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..12.

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

EXAMPLE

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}}

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<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}]

CROSSREFS

Crossing set partitions are A016098.

Non-capturing set partitions are A326254.

Crossing, capturing set partitions are A326246.

Cf. A000108, A000110, A001519, A054391, A058681, A095661, A324170.

Cf. A326212, A326237, A326243, A326248, A326249.

Sequence in context: A055852 A319405 A122611 * A219756 A014915 A137747

Adjacent sequences:  A326242 A326243 A326244 * A326246 A326247 A326248

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jun 20 2019

STATUS

approved

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Last modified July 24 03:11 EDT 2021. Contains 346271 sequences. (Running on oeis4.)