cp's OEIS Frontend

A317079 Number of unlabeled antichains of multisets with multiset-join a multiset of size n.

%I A317079 #7 Jul 20 2018 22:30:22
%S A317079 1,1,3,9,46,450
%N A317079 Number of unlabeled antichains of multisets with multiset-join a multiset of size n.
%C A317079 An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. The multiset-join of a multiset system has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.
%H A317079 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 A317079 Non-isomorphic representatives of the a(3) = 9 antichains of multisets:
%e A317079   (111),
%e A317079   (122), (1)(22), (12)(22),
%e A317079   (123), (1)(23), (13)(23), (1)(2)(3), (12)(13)(23).
%t A317079 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 A317079 multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}]
%t A317079 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A317079 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A317079 auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],multijoin@@#==m&];
%t A317079 sysnorm[m_]:=First[Sort[sysnorm[m,1]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
%t A317079 Table[Length[Union[sysnorm/@Join@@Table[auu[m],{m,strnorm[n]}]]],{n,5}]
%Y A317079 Cf. A007716, A255906, A261006, A285572, A293993, A293994, A304998.
%Y A317079 Cf. A317073, A317074, A317075, A317076, A317080.
%K A317079 nonn,more
%O A317079 0,3
%A A317079 _Gus Wiseman_, Jul 20 2018