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A326336 Number of set partitions of {1..n} whose capturing blocks are connected. 5
1, 1, 1, 1, 2, 7, 24, 100, 458, 2279, 12270 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A123764 A150437 A150438 * A150439 A150440 A150441
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 28 2019
STATUS
approved

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Last modified July 23 23:47 EDT 2024. Contains 374575 sequences. (Running on oeis4.)