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A317073 Number of antichains of multisets with multiset-join a normal multiset of size n. 9
1, 1, 3, 16, 198, 9890 (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 normal if it spans an initial interval of positive integers. The multiset-join of a set of multisets 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) = 16 antichains of multisets:


  (122), (12)(22), (1)(22),

  (112), (11)(12), (2)(11),

  (123), (13)(23), (12)(23), (12)(13), (12)(13)(23), (3)(12), (2)(13), (1)(23), (1)(2)(3).


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}]]];

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

auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]], submultisetQ], multijoin@@#==m&];

Table[Length[Join@@Table[auu[m], {m, allnorm[n]}]], {n, 5}]


Cf. A048143, A255906, A285572, A303837, A303838, A304998, A305001.

Cf. A317074, A317075, A317077, A317079.

Sequence in context: A174137 A166860 A196562 * A272658 A326903 A113597

Adjacent sequences:  A317070 A317071 A317072 * A317074 A317075 A317076




Gus Wiseman, Jul 20 2018



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Last modified January 22 04:27 EST 2020. Contains 331133 sequences. (Running on oeis4.)