A326572 -id:A326572 - cp's OEIS frontend

A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

2, 3, 5, 10, 22, 61, 247, 2096, 81896, 52260575

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(0) = 2 through a(4) = 22 antichains:
  {}    {}     {}       {}           {}
  {{}}  {{}}   {{}}     {{}}         {{}}
        {{1}}  {{1}}    {{1}}        {{1}}
               {{2}}    {{2}}        {{2}}
               {{1,2}}  {{3}}        {{3}}
                        {{1,2}}      {{4}}
                        {{1,3}}      {{1,2}}
                        {{2,3}}      {{1,3}}
                        {{1,2,3}}    {{1,4}}
                        {{3},{1,2}}  {{2,3}}
                                     {{2,4}}
                                     {{3,4}}
                                     {{1,2,3}}
                                     {{1,2,4}}
                                     {{1,3,4}}
                                     {{2,3,4}}
                                     {{1,2,3,4}}
                                     {{3},{1,2}}
                                     {{4},{1,3}}
                                     {{1,4},{2,3}}
                                     {{2,4},{1,2,3}}
                                     {{3,4},{1,2,4}}

Set partitions with equal block-sums are A035470.
Antichains with different edge-sums are A326030.
MM-numbers of multiset partitions with equal part-sums are A326534.
The covering case is A326566.
Cf. A000372, A003182, A006126, A307249, A321455, A321717, A321718, A326518, A326565, A326572.

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]]]];
cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&];
Table[Length[cleqset[Range[n]]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 13 2019

A326571 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having different sums.

Original entry on oeis.org

1, 0, 1, 5, 61, 2721, 788221

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(3) = 5 antichains:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 61 antichains:
  {1234}  {12}{34}    {12}{13}{14}   {12}{13}{14}{24}   {12}{13}{14}{24}{34}
          {13}{24}    {12}{13}{24}   {12}{13}{14}{34}   {12}{13}{23}{24}{34}
          {12}{134}   {12}{13}{34}   {12}{13}{23}{24}
          {12}{234}   {12}{14}{34}   {12}{13}{23}{34}
          {13}{124}   {12}{23}{24}   {12}{13}{24}{34}
          {13}{234}   {12}{23}{34}   {12}{14}{24}{34}
          {14}{123}   {12}{24}{34}   {12}{23}{24}{34}
          {14}{234}   {13}{14}{24}   {13}{14}{24}{34}
          {23}{124}   {13}{23}{24}   {13}{23}{24}{34}
          {23}{134}   {13}{23}{34}   {12}{13}{14}{234}
          {24}{134}   {13}{24}{34}   {12}{23}{24}{134}
          {34}{123}   {14}{24}{34}   {123}{124}{134}{234}
          {123}{124}  {12}{13}{234}
          {123}{134}  {12}{14}{234}
          {123}{234}  {12}{23}{134}
          {124}{134}  {12}{24}{134}
          {124}{234}  {13}{14}{234}
          {134}{234}  {13}{23}{124}
                      {14}{34}{123}
                      {23}{24}{134}
                      {12}{134}{234}
                      {13}{124}{234}
                      {14}{123}{234}
                      {23}{124}{134}
                      {123}{124}{134}
                      {123}{124}{234}
                      {123}{134}{234}
                      {124}{134}{234}

Antichain covers are A006126.
Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
Antichain covers with equal edge-sums and no singletons are A326565.
Antichain covers with different edge-sizes and no singletons are A326569.
The case with singletons allowed is A326572.
Antichains with equal edge-sums are A326574.
Cf. A000372, A003182, A035470, A307249, A321469, A326519, A326566, A326570, A326573.

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]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n],{2,n}],SubsetQ[#1,#2]||Total[#1]==Total[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,5}]

A326573 Number of connected antichains of subsets of {1..n}, all having different sums.

Original entry on oeis.org

1, 1, 1, 5, 59, 2689, 787382

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(3) = 5 antichains:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 59 antichains:
  {1234}  {12}{134}   {12}{13}{14}   {12}{13}{14}{24}   {12}{13}{14}{24}{34}
          {12}{234}   {12}{13}{24}   {12}{13}{14}{34}   {12}{13}{23}{24}{34}
          {13}{124}   {12}{13}{34}   {12}{13}{23}{24}
          {13}{234}   {12}{14}{34}   {12}{13}{23}{34}
          {14}{123}   {12}{23}{24}   {12}{13}{24}{34}
          {14}{234}   {12}{23}{34}   {12}{14}{24}{34}
          {23}{124}   {12}{24}{34}   {12}{23}{24}{34}
          {23}{134}   {13}{14}{24}   {13}{14}{24}{34}
          {24}{134}   {13}{23}{24}   {13}{23}{24}{34}
          {34}{123}   {13}{23}{34}   {12}{13}{14}{234}
          {123}{124}  {13}{24}{34}   {12}{23}{24}{134}
          {123}{134}  {14}{24}{34}   {123}{124}{134}{234}
          {123}{234}  {12}{13}{234}
          {124}{134}  {12}{14}{234}
          {124}{234}  {12}{23}{134}
          {134}{234}  {12}{24}{134}
                      {13}{14}{234}
                      {13}{23}{124}
                      {14}{34}{123}
                      {23}{24}{134}
                      {12}{134}{234}
                      {13}{124}{234}
                      {14}{123}{234}
                      {23}{124}{134}
                      {123}{124}{134}
                      {123}{124}{234}
                      {123}{134}{234}
                      {124}{134}{234}

Antichain covers are A006126.
Connected antichains are A048143.
Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
Antichain covers with equal edge-sums are A326566.
The non-connected case is A326572.
Cf. A000372, A293510, A307249, A321469, A323818, A326519, A326565, A326569, A326570, A326571.

A326030 Number of antichains of subsets of {1..n} with different edge-sums.

Original entry on oeis.org

2, 3, 6, 19, 132, 3578, 826949

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(0) = 2 through a(3) = 19 antichains:
  {}    {}     {}         {}
  {{}}  {{}}   {{}}       {{}}
        {{1}}  {{1}}      {{1}}
               {{2}}      {{2}}
               {{1,2}}    {{3}}
               {{1},{2}}  {{1,2}}
                          {{1,3}}
                          {{2,3}}
                          {{1},{2}}
                          {{1,2,3}}
                          {{1},{3}}
                          {{2},{3}}
                          {{1},{2,3}}
                          {{2},{1,3}}
                          {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
The covering case is A326572.
Antichains with equal edge-sums are A326574.
Cf. A000372, A003182, A006126, A321469, A326519, A326571, A326573.

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]]]];
cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]==Total[#2]&];
Table[Length[cleqset[Range[n]]],{n,0,5}]

A326569 Number of covering antichains of subsets of {1..n} with no singletons and different edge-sizes.

Original entry on oeis.org

1, 0, 1, 1, 13, 121, 2566, 121199, 13254529

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sizes are the numbers of vertices in each edge, so for example the edge sizes of {{1,3},{2,5},{3,4,5}} are {2,2,3}.

			The a(2) = 1 through a(4) = 13 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}
                      {{1,2},{1,3,4}}
                      {{1,2},{2,3,4}}
                      {{1,3},{1,2,4}}
                      {{1,3},{2,3,4}}
                      {{1,4},{1,2,3}}
                      {{1,4},{2,3,4}}
                      {{2,3},{1,2,4}}
                      {{2,3},{1,3,4}}
                      {{2,4},{1,2,3}}
                      {{2,4},{1,3,4}}
                      {{3,4},{1,2,3}}
                      {{3,4},{1,2,4}}

Antichain covers are A006126.
Set partitions with different block sizes are A007837.
The case with singletons is A326570.
Cf. A000372, A003182, A306021, A307249, A326565, A326571, A326572, A326573.

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]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n],{2,n}],SubsetQ[#1,#2]||Length[#1]==Length[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,6}]

a(n) = A326570(n) - n*a(n-1) for n > 0. - Andrew Howroyd, Aug 13 2019

a(8) from Andrew Howroyd, Aug 13 2019

A326570 Number of covering antichains of subsets of {1..n} with different edge-sizes.

Original entry on oeis.org

2, 1, 1, 4, 17, 186, 3292, 139161, 14224121

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sizes are the numbers of vertices in each edge, so for example the edge-sizes of {{1,3},{2,5},{3,4,5}} are {2,2,3}.

			The a(0) = 2 through a(4) = 17 antichains:
  {}    {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}
  {{}}                  {{1},{2,3}}  {{1},{2,3,4}}
                        {{2},{1,3}}  {{2},{1,3,4}}
                        {{3},{1,2}}  {{3},{1,2,4}}
                                     {{4},{1,2,3}}
                                     {{1,2},{1,3,4}}
                                     {{1,2},{2,3,4}}
                                     {{1,3},{1,2,4}}
                                     {{1,3},{2,3,4}}
                                     {{1,4},{1,2,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,4}}
                                     {{2,3},{1,3,4}}
                                     {{2,4},{1,2,3}}
                                     {{2,4},{1,3,4}}
                                     {{3,4},{1,2,3}}
                                     {{3,4},{1,2,4}}

Antichain covers are A006126.
Set partitions with different block sizes are A007837.
The case without singletons is A326569.
(Antichain) covers with equal edge-sizes are A306021.
Cf. A000372, A003182, A307249, A326026, A326514, A326566, A326572.

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]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n]],SubsetQ[#1,#2]||Length[#1]==Length[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,6}]

a(8) from Andrew Howroyd, Aug 13 2019

A327903 Number of set-systems covering n vertices where every edge has a different sum.

Original entry on oeis.org

1, 1, 5, 77, 7369, 10561753, 839653402893, 15924566366443524837, 315320784127456186118309342769, 29238175285109256786706269143580213236526609, 59347643832090275881798554403880633753161146711444051797893301

Offset: 0

Gus Wiseman, Sep 30 2019

nonn

A set-system is a set of nonempty sets. It is covering if there are no isolated (uncovered) vertices.

			The a(3) = 77 set-systems:
  123  1-23    1-2-3      1-2-3-13      1-2-3-13-23     1-2-3-13-23-123
       2-13    1-2-13     1-2-3-23      1-2-12-13-23    1-2-12-13-23-123
       1-123   1-2-23     1-2-12-13     1-2-3-13-123
       12-13   1-3-23     1-2-12-23     1-2-3-23-123
       12-23   2-3-13     1-2-13-23     1-2-12-13-123
       13-23   1-12-13    1-2-3-123     1-2-12-23-123
       2-123   1-12-23    1-3-13-23     1-2-13-23-123
       3-123   1-13-23    2-3-13-23     1-3-13-23-123
       12-123  1-2-123    1-12-13-23    2-3-13-23-123
       13-123  1-3-123    1-2-12-123    1-12-13-23-123
       23-123  2-12-13    1-2-13-123    2-12-13-23-123
               2-12-23    1-2-23-123
               2-13-23    1-3-13-123
               2-3-123    1-3-23-123
               3-13-23    2-12-13-23
               1-12-123   2-3-13-123
               1-13-123   2-3-23-123
               12-13-23   1-12-13-123
               1-23-123   1-12-23-123
               2-12-123   1-13-23-123
               2-13-123   2-12-13-123
               2-23-123   2-12-23-123
               3-13-123   2-13-23-123
               3-23-123   3-13-23-123
               12-13-123  12-13-23-123
               12-23-123
               13-23-123

Andrew Howroyd, Table of n, a(n) for n = 0..20
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The antichain case is A326572.
The graphical case is A327904.
Cf. A003465, A006126, A035470, A275780, A321469, A326030, A326519, A326535, A326565, A326571, A326573.

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]]]];
qes[n_]:=Select[stableSets[Subsets[Range[n],{1,n}],Total[#1]==Total[#2]&],Union@@#==Range[n]&];
Table[Length[qes[n]],{n,0,4}]

\\ by inclusion/exclusion on covered vertices.
C(v)={my(u=Vecrev(-1 + prod(k=1, #v, 1 + x^v[k]))); prod(i=1, #u, 1 + u[i])}
a(n)={my(s=0); forsubset(n, v, s += (-1)^(n-#v)*C(v)); s} \\ Andrew Howroyd, Oct 02 2019

Terms a(4) and beyond from Andrew Howroyd, Oct 02 2019

cp's OEIS Frontend

A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

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A326571 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having different sums.

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A326573 Number of connected antichains of subsets of {1..n}, all having different sums.

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A326030 Number of antichains of subsets of {1..n} with different edge-sums.

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A326569 Number of covering antichains of subsets of {1..n} with no singletons and different edge-sizes.

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A326570 Number of covering antichains of subsets of {1..n} with different edge-sizes.

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A327903 Number of set-systems covering n vertices where every edge has a different sum.

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