A326336 Number of set partitions of {1..n} whose capturing blocks are connected.

1, 1, 1, 1, 2, 7, 24, 100, 458, 2279, 12270

Offset

0,5

Comments

Two blocks are capturing 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 capturing blocks connected if the graph whose vertices are the blocks and whose edges are capturing pairs of blocks is connected.

Links

Table of n, a(n) for n=0..10.

Example

The a(0) = 1 through a(6) = 24 set partitions:
  {}  {1}  {12}  {123}  {1234}    {12345}    {123456}
                        {14}{23}  {125}{34}  {1236}{45}
                                  {134}{25}  {1245}{36}
                                  {135}{24}  {1246}{35}
                                  {14}{235}  {125}{346}
                                  {145}{23}  {1256}{34}
                                  {15}{234}  {126}{345}
                                             {134}{256}
                                             {1345}{26}
                                             {1346}{25}
                                             {135}{246}
                                             {1356}{24}
                                             {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}

Mathematica

capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<t<y||z<x<y<t];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
captcmpts[stn_]:=csm[Union[List/@stn, Select[Subsets[stn, {2}], capXQ]]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[sps[Range[n]], Length[captcmpts[#]]<=1&]], {n, 0, 6}]

Crossrefs

Simple graphs whose capturing blocks are connected are A326330.

Set partitions whose crossing blocks are connected are A099947.

Set partitions whose nesting blocks are connected are A326335.

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

Sequence in context: A123764 A150437 A150438 * A150439 A150440 A150441

Adjacent sequences: A326333 A326334 A326335 * A326337 A326338 A326339

Keyword

nonn,more

Author

Gus Wiseman, Jun 28 2019

Status

approved