A326335 Number of set partitions of {1..n} whose nesting blocks are connected.

1, 1, 1, 1, 2, 6, 21, 86, 394, 1974, 10696

Offset: 0

Author

Gus Wiseman, Jun 27 2019

Keywords

• nonn
• more

Comments

Two blocks are nesting if they are of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. A set partition has its nesting blocks connected if the graph whose vertices are the blocks and whose edges are nesting pairs of blocks is connected.

Examples

			The a(0) = 1 through a(6) = 21 set partitions:
  {}  {1}  {12}  {123}  {1234}    {12345}    {123456}
                        {14}{23}  {125}{34}  {1236}{45}
                                  {134}{25}  {1245}{36}
                                  {14}{235}  {125}{346}
                                  {145}{23}  {1256}{34}
                                  {15}{234}  {126}{345}
                                             {134}{256}
                                             {1345}{26}
                                             {1346}{25}
                                             {136}{245}
                                             {14}{2356}
                                             {145}{236}
                                             {1456}{23}
                                             {146}{235}
                                             {15}{2346}
                                             {156}{234}
                                             {16}{2345}
                                             {15}{26}{34}
                                             {16}{23}{45}
                                             {16}{24}{35}
                                             {16}{25}{34}
		

Crossrefs

Simple graphs whose nesting blocks are connected are A326330.

Set partitions whose crossing blocks are connected are A099947.

Set partitions whose capturing blocks are connected are A326336.

Cf. A000110, A001519, A016098, A122880, A324173, A326243, A326248, A326293, A326331, A326337.

Programs

Mathematica

nesXQ[stn_]:=MatchQ[stn,{_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
nestcmpts[stn_]:=csm[Union[List/@stn,Select[Subsets[stn,{2}],nesXQ]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Length[nestcmpts[#]]<=1&]],{n,0,5}]