cp's OEIS Frontend

A320356 Number of strict connected antichains of multisets whose multiset union is an integer partition of n.

%I A320356 #6 Oct 12 2018 22:43:20
%S A320356 1,1,2,3,5,8,13,22,35,62,98,171,277
%N A320356 Number of strict connected antichains of multisets whose multiset union is an integer partition of n.
%H A320356 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 A320356 The a(1) = 1 through a(6) = 13 clutters:
%e A320356   {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
%e A320356          {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
%e A320356                   {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
%e A320356                              {{1,1,2}}    {{1,1,3}}      {{3,3}}
%e A320356                              {{1,1,1,1}}  {{1,2,2}}      {{1,1,4}}
%e A320356                                           {{1,1,1,2}}    {{1,2,3}}
%e A320356                                           {{1,1,1,1,1}}  {{2,2,2}}
%e A320356                                           {{1,1},{1,2}}  {{1,1,1,3}}
%e A320356                                                          {{1,1,2,2}}
%e A320356                                                          {{1,1,1,1,2}}
%e A320356                                                          {{1,1},{1,3}}
%e A320356                                                          {{1,1,1,1,1,1}}
%e A320356                                                          {{1,2},{1,1,1}}
%t A320356 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320356 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320356 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A320356 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A320356 antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
%t A320356 Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]
%Y A320356 Cf. A001970, A007718, A048143, A056156, A258466, A261006, A293994, A318403, A319079, A319719, A319721, A320351, A320353, A320355.
%K A320356 nonn,more
%O A320356 0,3
%A A320356 _Gus Wiseman_, Oct 11 2018