A317075 - OEIS

%I #7 Jul 20 2018 22:29:55

%S 1,1,2,10,147,8998

%N Number of connected antichains of multisets with multiset-join a 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 normal if it spans an initial interval of positive integers. 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) = 10 connected antichains of multisets:

%e   (111),

%e   (122), (12)(22),

%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 allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];

%t cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],And[multijoin@@#==m,Length[csm[#]]==1]&];

%t Table[Length[Join@@Table[cuu[m],{m,allnorm[n]}]],{n,5}]

%Y Cf. A048143, A007718, A255906, A286520, A303837, A303838, A304716, A305001, A305078.

%Y Cf. A317073, A317076, A317077, A317080.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jul 20 2018