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

0,5

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.

Links

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

Example

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}

Mathematica

nesXQ[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]]]]]]]]];
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}]

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.

Sequence in context: A344229 A090805 A150226 * A256180 A150227 A263852

Adjacent sequences: A326332 A326333 A326334 * A326336 A326337 A326338

Keyword

nonn,more

Author

Gus Wiseman, Jun 27 2019

Status

approved