A320449 Number of antichains of sets whose multiset union is an integer partition of n.

1, 1, 2, 4, 6, 9, 18, 24, 39, 58, 92, 131, 206

Offset

0,3

Links

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

Example

The a(1) = 1 through a(7) = 24 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1},{1}}  {{1,2}}        {{1,3}}            {{1,4}}
                    {{1},{2}}      {{1},{3}}          {{2,3}}
                    {{1},{1},{1}}  {{2},{2}}          {{1},{4}}
                                   {{1},{1},{2}}      {{2},{3}}
                                   {{1},{1},{1},{1}}  {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}
.
  {{6}}                      {{7}}
  {{1,5}}                    {{1,6}}
  {{2,4}}                    {{2,5}}
  {{1,2,3}}                  {{3,4}}
  {{1},{5}}                  {{1,2,4}}
  {{2},{4}}                  {{1},{6}}
  {{3},{3}}                  {{2},{5}}
  {{1},{2,3}}                {{3},{4}}
  {{2},{1,3}}                {{1},{2,4}}
  {{3},{1,2}}                {{2},{1,4}}
  {{1},{1},{4}}              {{4},{1,2}}
  {{1,2},{1,2}}              {{1},{1},{5}}
  {{1},{2},{3}}              {{1,2},{1,3}}
  {{2},{2},{2}}              {{1},{2},{4}}
  {{1},{1},{1},{3}}          {{1},{3},{3}}
  {{1},{1},{2},{2}}          {{2},{2},{3}}
  {{1},{1},{1},{1},{2}}      {{1},{1},{2,3}}
  {{1},{1},{1},{1},{1},{1}}  {{1},{1},{1},{4}}
                             {{1},{1},{2},{3}}
                             {{1},{2},{2},{2}}
                             {{1},{1},{1},{1},{3}}
                             {{1},{1},{1},{2},{2}}
                             {{1},{1},{1},{1},{1},{2}}
                             {{1},{1},{1},{1},{1},{1},{1}}

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]]]];
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[And@@UnsameQ@@@#, antiQ[#]]&]], {n, 10}]

Crossrefs

Cf. A001970, A089259, A258466, A319719, A319721, A320328, A320353, A320355, A320356.

Sequence in context: A244470 A098787 A164138 * A263434 A192079 A079143

Adjacent sequences: A320446 A320447 A320448 * A320450 A320451 A320452

Keyword

nonn,more

Author

Gus Wiseman, Oct 12 2018

Status

approved