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

Author

Gus Wiseman, Jun 20 2019

Keywords

• nonn
• more

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.

Examples

			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}}
		

Links

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

Crossrefs

Crossing set partitions are A016098.

Non-capturing set partitions are A054391.

Crossing, capturing set partitions are A326246.

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

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

Programs

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_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x