A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.

1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 13, 22, 31

Offset

0,4

Links

Table of n, a(n) for n=0..12.

Example

The a(1) = 1 through a(10) = 13 clutters:
  {{1}}  {{2}}  {{3}}    {{4}}    {{5}}    {{6}}      {{7}}
                {{1,2}}  {{1,3}}  {{1,4}}  {{1,5}}    {{1,6}}
                                  {{2,3}}  {{2,4}}    {{2,5}}
                                           {{1,2,3}}  {{3,4}}
                                                      {{1,2,4}}
                                                      {{1,2},{1,3}}
.
  {{8}}          {{9}}          {{10}}
  {{1,7}}        {{1,8}}        {{1,9}}
  {{2,6}}        {{2,7}}        {{2,8}}
  {{3,5}}        {{3,6}}        {{3,7}}
  {{1,2,5}}      {{4,5}}        {{4,6}}
  {{1,3,4}}      {{1,2,6}}      {{1,2,7}}
  {{1,2},{1,4}}  {{1,3,5}}      {{1,3,6}}
  {{1,2},{2,3}}  {{2,3,4}}      {{1,4,5}}
                 {{1,2},{1,5}}  {{2,3,5}}
                 {{1,2},{2,4}}  {{1,2,3,4}}
                 {{1,3},{1,4}}  {{1,2},{1,6}}
                 {{1,3},{2,3}}  {{1,2},{2,5}}
                                {{1,3},{1,5}}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_, N_]:=Or[Length[M]==0, MatchQ[{Sort[List@@M], Sort[List@@N]}, {{x_, Z___}, {___, x_, W___}}/; submultisetQ[{Z}, {W}]]];
antiQ[s_]:=Select[Tuples[s, 2], And[UnsameQ@@#, submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n], And[UnsameQ@@#, And@@UnsameQ@@@#, Length[csm[#]]==1, antiQ[#]]&]], {n, 8}]

Crossrefs

Cf. A001970, A007718, A048143, A050342, A056156, A089259, A261049, A293994, A319719, A320351, A320353, A320355, A320356.

Sequence in context: A214076 A067859 A006207 * A362048 A334626 A339235

Adjacent sequences: A318400 A318401 A318402 * A318404 A318405 A318406

Keyword

nonn,more

Author

Gus Wiseman, Oct 12 2018

Status

approved