A320353 Number of antichains of multisets whose multiset union is an integer partition of n.

1, 1, 3, 5, 11, 17, 36, 56, 107, 175, 311, 505, 887

Offset

0,3

Links

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

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Example

The a(1) = 1 through a(5) = 17 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{2},{1,1}}        {{1,1,1,2}}
                                   {{1,1},{1,1}}      {{1},{2,2}}
                                   {{1},{1},{2}}      {{3},{1,1}}
                                   {{1},{1},{1},{1}}  {{1,1,1,1,1}}
                                                      {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{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], antiQ]], {n, 8}]

Crossrefs

Cf. A001970, A006126, A050342, A089259, A258466, A261005, A261049, A293994, A319719, A319721, A320328, A320355, A320356.

Sequence in context: A261674 A319632 A060647 * A155989 A125557 A007455

Adjacent sequences: A320350 A320351 A320352 * A320354 A320355 A320356

Keyword

nonn,more

Author

Gus Wiseman, Oct 11 2018

Status

approved