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A320449 Number of antichains of sets whose multiset union is an integer partition of n. 8
1, 1, 2, 4, 6, 9, 18, 24, 39, 58, 92, 131, 206 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 1 02:15 EDT 2020. Contains 333153 sequences. (Running on oeis4.)