A317080 Number of unlabeled connected antichains of multisets with multiset-join a multiset of size n.

1, 1, 2, 6, 34, 392

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. 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

Non-isomorphic representatives of the a(3) = 6 connected antichains of multisets:
  (111),
  (122), (12)(22),
  (123), (13)(23), (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]]]]]]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
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]]}]])]];
cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]], submultisetQ], And[multijoin@@#==m, Length[csm[#]]==1]&];
Table[Length[Union[sysnorm/@Join@@Table[cuu[m], {m, strnorm[n]}]]], {n, 5}]

Crossrefs

Cf. A007716, A007718, A048143, A261006, A286520, A293993, A293994, A304716, A304998.

Cf. A317073, A317074, A317075, A317076, A317077, A317078, A317079.

Sequence in context: A191742 A181082 A118186 * A075272 A353536 A224913

Adjacent sequences: A317077 A317078 A317079 * A317081 A317082 A317083

Keyword

nonn,more

Author

Gus Wiseman, Jul 20 2018

Status

approved