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A317076 Number of connected antichains of multisets with multiset-join a strongly normal multiset of size n. 5
1, 1, 2, 8, 110, 7047 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=0..5.

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).


The a(3) = 8 connected antichains of multisets:


(112), (11)(12),

(123), (13)(23), (12)(23), (12)(13), (12)(13)(23).


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]]]];

multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}];

submultisetQ[M_, N_]:=Or[Length[M]==0, MatchQ[{Sort[List@@M], Sort[List@@N]}, {{x_, Z___}, {___, x_, W___}}/; submultisetQ[{Z}, {W}]]];

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]]]]]]]]];

strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];

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

Table[Length[Join@@Table[cuu[m], {m, strnorm[n]}]], {n, 5}]


Cf. A048143, A255906, A286520, A303837, A303838, A304716, A305001, A305078.

Cf. A317074, A317075, A317078, A317080.

Sequence in context: A132523 A012554 A012665 * A211938 A331405 A005787

Adjacent sequences: A317073 A317074 A317075 * A317077 A317078 A317079




Gus Wiseman, Jul 20 2018



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Last modified December 2 05:20 EST 2022. Contains 358485 sequences. (Running on oeis4.)