A319719 -id:A319719 - cp's OEIS frontend

A321184 Number of integer partitions of n that are the vertex-degrees of some multiset of nonempty sets, none of which is a proper subset of any other, with no singletons.

1, 0, 1, 1, 3, 2, 7, 6, 15, 15, 30

Offset: 0

Gus Wiseman, Oct 29 2018

			The a(2) = 1 through a(8) = 15 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (211)   (11111)  (222)     (3211)     (332)
               (1111)           (321)     (22111)    (422)
                                (2211)    (31111)    (431)
                                (3111)    (211111)   (2222)
                                (21111)   (1111111)  (3221)
                                (111111)             (3311)
                                                     (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(6) = 7 integer partitions together with a realizing multi-antichain of each (the parts of the partition count the appearances of each vertex in the multi-antichain):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2,3},{1,2,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}

Cf. A000070, A000569, A006126, A096827, A147878, A209816, A283877, A318360, A319719, A319721, A320799, A320921, A321176.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multanti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multanti[#]!={}&]],{n,8}]

A321678 Number of non-isomorphic weight-n strict antichains of sets with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 11, 13, 39, 67, 174

Offset: 0

Gus Wiseman, Nov 16 2018

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 11 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}    {{1,2,3,4,5,6}}
                      {{1,2},{3,4}}  {{1,2},{3,4,5}}  {{1,2},{3,4,5,6}}
                      {{1,3},{2,3}}  {{1,4},{2,3,4}}  {{1,2,3},{4,5,6}}
                                                      {{1,2,5},{3,4,5}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{1,5},{2,3,4,5}}
                                                      {{1,2},{1,3},{2,3}}
                                                      {{1,2},{3,4},{5,6}}
                                                      {{1,2},{3,5},{4,5}}
                                                      {{1,3},{2,4},{3,4}}
                                                      {{1,4},{2,4},{3,4}}

Cf. A006126, A049311, A096827, A285572, A293993, A293994, A318099, A319719, A319721, A320799, A321679.

A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 41, 70, 123, 208, 355, 597

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(6) = 23 antichains:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
                  {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
                  {{1},{2}}  {{1,1,2}}    {{1,1,3}}      {{3,3}}
                             {{1},{3}}    {{1,2,2}}      {{1,1,4}}
                             {{1,1,1,1}}  {{1},{4}}      {{1,2,3}}
                             {{2},{1,1}}  {{2},{3}}      {{1},{5}}
                                          {{1,1,1,2}}    {{2,2,2}}
                                          {{1},{2,2}}    {{2},{4}}
                                          {{3},{1,1}}    {{1,1,1,3}}
                                          {{1,1,1,1,1}}  {{1,1,2,2}}
                                          {{1,1},{1,2}}  {{1},{2,3}}
                                          {{2},{1,1,1}}  {{2},{1,3}}
                                                         {{3},{1,2}}
                                                         {{4},{1,1}}
                                                         {{1,1,1,1,2}}
                                                         {{1,1},{1,3}}
                                                         {{1,1},{2,2}}
                                                         {{1},{2},{3}}
                                                         {{3},{1,1,1}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1,2},{1,1,1}}
                                                         {{2},{1,1,1,1}}

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

Cf. A001970, A258466, A261049, A319719, A319721, A320353, A320355, A320356.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,antiQ[#]]&]],{n,10}]

A320450 Number of strict antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 10, 13, 19, 28, 47, 64, 98

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(8) = 19 antichains:
  {{1}}  {{2}}  {{3}}      {{4}}      {{5}}      {{6}}
                {{1,2}}    {{1,3}}    {{1,4}}    {{1,5}}
                {{1},{2}}  {{1},{3}}  {{2,3}}    {{2,4}}
                                      {{1},{4}}  {{1,2,3}}
                                      {{2},{3}}  {{1},{5}}
                                                 {{2},{4}}
                                                 {{1},{2,3}}
                                                 {{2},{1,3}}
                                                 {{3},{1,2}}
                                                 {{1},{2},{3}}
.
  {{7}}          {{8}}
  {{1,6}}        {{1,7}}
  {{2,5}}        {{2,6}}
  {{3,4}}        {{3,5}}
  {{1,2,4}}      {{1,2,5}}
  {{1},{6}}      {{1,3,4}}
  {{2},{5}}      {{1},{7}}
  {{3},{4}}      {{2},{6}}
  {{1},{2,4}}    {{3},{5}}
  {{2},{1,4}}    {{1},{2,5}}
  {{4},{1,2}}    {{1},{3,4}}
  {{1,2},{1,3}}  {{2},{1,5}}
  {{1},{2},{4}}  {{3},{1,4}}
                 {{4},{1,3}}
                 {{5},{1,2}}
                 {{1,2},{1,4}}
                 {{1,2},{2,3}}
                 {{1},{2},{5}}
                 {{1},{3},{4}}

Cf. A001970, A050342, A089259, A258466, A261049, A319719, A319721, A320328, A320331, A320353.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]

A321484 Number of non-isomorphic self-dual connected multiset partitions of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 20, 35, 78, 141

Offset: 0

Gus Wiseman, Nov 16 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 9 multiset partitions:
  {{1}}  {{11}}  {{111}}    {{1111}}    {{11111}}      {{111111}}
                 {{2}{12}}  {{12}{12}}  {{11}{122}}    {{112}{122}}
                            {{2}{122}}  {{12}{122}}    {{12}{1222}}
                                        {{2}{1222}}    {{2}{12222}}
                                        {{2}{13}{23}}  {{22}{1122}}
                                        {{3}{3}{123}}  {{12}{13}{23}}
                                                       {{2}{13}{233}}
                                                       {{3}{23}{123}}
                                                       {{3}{3}{1233}}

Cf. A007718, A056156, A138178, A316983, A319565, A319616, A319647, A319719, A321194, A321585, A321680, A321681.

cp's OEIS Frontend

A321184 Number of integer partitions of n that are the vertex-degrees of some multiset of nonempty sets, none of which is a proper subset of any other, with no singletons.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A321678 Number of non-isomorphic weight-n strict antichains of sets with no singletons.

Views

Author

Keywords

Comments

Examples

Crossrefs

A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A320450 Number of strict antichains of sets whose multiset union is an integer partition of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A321484 Number of non-isomorphic self-dual connected multiset partitions of weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs