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%I #6 Jul 20 2018 22:30:04
%S 1,1,2,8,110,7047
%N Number of connected antichains of multisets with multiset-join a strongly normal multiset of size n.
%C An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. The multiset-join of a multiset system has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.
%H 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 The a(3) = 8 connected antichains of multisets:
%e (111),
%e (112), (11)(12),
%e (123), (13)(23), (12)(23), (12)(13), (12)(13)(23).
%t stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
%t multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
%t submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t 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]]],multijoin@@s[[c[[1]]]]]]]]];
%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],And[multijoin@@#==m,Length[csm[#]]==1]&];
%t Table[Length[Join@@Table[cuu[m],{m,strnorm[n]}]],{n,5}]
%Y Cf. A048143, A255906, A286520, A303837, A303838, A304716, A305001, A305078.
%Y Cf. A317074, A317075, A317078, A317080.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jul 20 2018
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