A317075 Number of connected antichains of multisets with multiset-join a normal multiset of size n.

1, 1, 2, 10, 147, 8998

Offset

0,3

Comments

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.

Links

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

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

Example

The a(3) = 10 connected antichains of multisets:
  (111),
  (122), (12)(22),
  (112), (11)(12),
  (123), (13)(23), (12)(23), (12)(13), (12)(13)(23).

Mathematica

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]]]]]]]]];
allnorm[n_]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]], submultisetQ], And[multijoin@@#==m, Length[csm[#]]==1]&];
Table[Length[Join@@Table[cuu[m], {m, allnorm[n]}]], {n, 5}]

Crossrefs

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

Cf. A317073, A317076, A317077, A317080.

Sequence in context: A057565 A152804 A188490 * A295207 A213457 A060595

Adjacent sequences: A317072 A317073 A317074 * A317076 A317077 A317078

Keyword

nonn,more

Author

Gus Wiseman, Jul 20 2018

Status

approved