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A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n. 2
1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 13, 22, 31 (list; graph; refs; listen; history; text; internal format)
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 * A332755 A017912 A102543

Adjacent sequences:  A318400 A318401 A318402 * A318404 A318405 A318406

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Oct 12 2018

STATUS

approved

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Last modified July 9 13:27 EDT 2020. Contains 335543 sequences. (Running on oeis4.)