A326245 Number of crossing, non-capturing set partitions of {1..n}.

0, 0, 0, 0, 1, 7, 34, 141, 537, 1941, 6777, 23096, 77340

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