A326249 Number of capturing set partitions of {1..n} that are not nesting.

0, 0, 0, 0, 0, 1, 9, 55, 283, 1324, 5838, 24744

Offset: 0

Author

Gus Wiseman, Jun 20 2019

Keywords

• nonn
• more

Comments

Capturing is a weaker condition than nesting. A set partition is capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. For example, {{1,3,5},{2,4}} is capturing but not nesting, so is counted under a(5).

Examples

			The a(6) = 9 set partitions:
  {{1},{2,4,6},{3,5}}
  {{1,3,5},{2,4},{6}}
  {{1,3,6},{2,4},{5}}
  {{1,3,6},{2,5},{4}}
  {{1,4,6},{2},{3,5}}
  {{1,4,6},{2,5},{3}}
  {{1,3,5},{2,4,6}}
  {{1,2,4,6},{3,5}}
  {{1,3,5,6},{2,4}}
		

Crossrefs

MM-numbers of capturing, non-nesting multiset partitions are A326260.

Nesting set partitions are A016098.

Capturing set partitions are A326243.

Non-crossing, nesting set partitions are A122880 (conjectured).

Cf. A000110, A054391, A058681, A095661, A099947.

Cf. A326212, A326237, A326245, A326246, A054391, A326255, A326256.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
capXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;x