cp's OEIS Frontend

A317074 Number of antichains of multisets with multiset-join a strongly normal multiset of size n.

%I A317074 #6 Jul 20 2018 22:29:48
%S A317074 1,1,3,13,148,7685
%N A317074 Number of antichains of multisets with multiset-join a strongly normal multiset of size n.
%C A317074 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 A317074 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 A317074 The a(3) = 13 antichains of multisets:
%e A317074   (111),
%e A317074   (112), (11)(12), (2)(11),
%e A317074   (123), (13)(23), (12)(23), (12)(13), (12)(13)(23), (3)(12), (2)(13), (1)(23), (1)(2)(3).
%t A317074 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 A317074 multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
%t A317074 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A317074 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A317074 auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],multijoin@@#==m&];
%t A317074 Table[Length[Join@@Table[auu[m],{m,strnorm[n]}]],{n,5}]
%Y A317074 Cf. A048143, A285572, A293993, A293994, A303837, A303838, A304998, A305001.
%Y A317074 Cf. A317073, A317076, A317078, A317079.
%K A317074 nonn,more
%O A317074 0,3
%A A317074 _Gus Wiseman_, Jul 20 2018