A326246 Number of crossing, capturing set partitions of {1..n}.

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

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<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, 5}]

Crossrefs

MM-numbers of crossing, capturing multiset partitions are A326259.

Crossing set partitions are A016098.

Capturing set partitions are A326243.

Crossing, nesting set partitions are A326248.

Crossing, non-capturing set partitions are A326245.

Non-crossing, capturing set partitions are A122880 (conjecture).

Cf. A000108, A000110, A058681, A099947, A324170, A326211, A326249, A054391, 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