cp's OEIS Frontend

A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

%I A319255 #15 Oct 13 2018 09:14:53
%S A319255 1,1,2,4,7,13,23,41,70,123,208,355,597
%N A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.
%H A319255 Goran Kilibarda and Vladeta Jovovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kilibarda/kili2.pdf">Antichains of Multisets</a>, Journal of Integer Sequences, Vol. 7 (2004).
%e A319255 The a(1) = 1 through a(6) = 23 antichains:
%e A319255   {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
%e A319255          {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
%e A319255                   {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
%e A319255                   {{1},{2}}  {{1,1,2}}    {{1,1,3}}      {{3,3}}
%e A319255                              {{1},{3}}    {{1,2,2}}      {{1,1,4}}
%e A319255                              {{1,1,1,1}}  {{1},{4}}      {{1,2,3}}
%e A319255                              {{2},{1,1}}  {{2},{3}}      {{1},{5}}
%e A319255                                           {{1,1,1,2}}    {{2,2,2}}
%e A319255                                           {{1},{2,2}}    {{2},{4}}
%e A319255                                           {{3},{1,1}}    {{1,1,1,3}}
%e A319255                                           {{1,1,1,1,1}}  {{1,1,2,2}}
%e A319255                                           {{1,1},{1,2}}  {{1},{2,3}}
%e A319255                                           {{2},{1,1,1}}  {{2},{1,3}}
%e A319255                                                          {{3},{1,2}}
%e A319255                                                          {{4},{1,1}}
%e A319255                                                          {{1,1,1,1,2}}
%e A319255                                                          {{1,1},{1,3}}
%e A319255                                                          {{1,1},{2,2}}
%e A319255                                                          {{1},{2},{3}}
%e A319255                                                          {{3},{1,1,1}}
%e A319255                                                          {{1,1,1,1,1,1}}
%e A319255                                                          {{1,2},{1,1,1}}
%e A319255                                                          {{2},{1,1,1,1}}
%t A319255 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A319255 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A319255 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A319255 antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
%t A319255 Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,antiQ[#]]&]],{n,10}]
%Y A319255 Cf. A001970, A258466, A261049, A319719, A319721, A320353, A320355, A320356.
%K A319255 nonn,more
%O A319255 0,3
%A A319255 _Gus Wiseman_, Oct 12 2018