cp's OEIS Frontend

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

%I A320449 #5 Oct 14 2018 09:16:30
%S A320449 1,1,2,4,6,9,18,24,39,58,92,131,206
%N A320449 Number of antichains of sets whose multiset union is an integer partition of n.
%e A320449 The a(1) = 1 through a(7) = 24 antichains:
%e A320449   {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
%e A320449          {{1},{1}}  {{1,2}}        {{1,3}}            {{1,4}}
%e A320449                     {{1},{2}}      {{1},{3}}          {{2,3}}
%e A320449                     {{1},{1},{1}}  {{2},{2}}          {{1},{4}}
%e A320449                                    {{1},{1},{2}}      {{2},{3}}
%e A320449                                    {{1},{1},{1},{1}}  {{1},{1},{3}}
%e A320449                                                       {{1},{2},{2}}
%e A320449                                                       {{1},{1},{1},{2}}
%e A320449                                                       {{1},{1},{1},{1},{1}}
%e A320449 .
%e A320449   {{6}}                      {{7}}
%e A320449   {{1,5}}                    {{1,6}}
%e A320449   {{2,4}}                    {{2,5}}
%e A320449   {{1,2,3}}                  {{3,4}}
%e A320449   {{1},{5}}                  {{1,2,4}}
%e A320449   {{2},{4}}                  {{1},{6}}
%e A320449   {{3},{3}}                  {{2},{5}}
%e A320449   {{1},{2,3}}                {{3},{4}}
%e A320449   {{2},{1,3}}                {{1},{2,4}}
%e A320449   {{3},{1,2}}                {{2},{1,4}}
%e A320449   {{1},{1},{4}}              {{4},{1,2}}
%e A320449   {{1,2},{1,2}}              {{1},{1},{5}}
%e A320449   {{1},{2},{3}}              {{1,2},{1,3}}
%e A320449   {{2},{2},{2}}              {{1},{2},{4}}
%e A320449   {{1},{1},{1},{3}}          {{1},{3},{3}}
%e A320449   {{1},{1},{2},{2}}          {{2},{2},{3}}
%e A320449   {{1},{1},{1},{1},{2}}      {{1},{1},{2,3}}
%e A320449   {{1},{1},{1},{1},{1},{1}}  {{1},{1},{1},{4}}
%e A320449                              {{1},{1},{2},{3}}
%e A320449                              {{1},{2},{2},{2}}
%e A320449                              {{1},{1},{1},{1},{3}}
%e A320449                              {{1},{1},{1},{2},{2}}
%e A320449                              {{1},{1},{1},{1},{1},{2}}
%e A320449                              {{1},{1},{1},{1},{1},{1},{1}}
%t A320449 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320449 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320449 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A320449 antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
%t A320449 Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]
%Y A320449 Cf. A001970, A089259, A258466, A319719, A319721, A320328, A320353, A320355, A320356.
%K A320449 nonn,more
%O A320449 0,3
%A A320449 _Gus Wiseman_, Oct 12 2018