cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A326566 Number of covering antichains of subsets of {1..n} with equal edge-sums.

Original entry on oeis.org

2, 1, 1, 2, 4, 14, 92, 1320, 73584, 51913039
Offset: 0

Views

Author

Gus Wiseman, Jul 13 2019

Keywords

Comments

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}.

Examples

			The a(1) = 1 through a(5) = 14 antichains:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}      {{1,2,3,4,5}}
                  {{3},{1,2}}  {{1,4},{2,3}}    {{1,2,5},{1,3,4}}
                               {{2,4},{1,2,3}}  {{1,3,5},{2,3,4}}
                               {{3,4},{1,2,4}}  {{1,4,5},{2,3,5}}
                                                {{5},{1,4},{2,3}}
                                                {{1,4,5},{1,2,3,4}}
                                                {{2,3,5},{1,2,3,4}}
                                                {{2,4,5},{1,2,3,5}}
                                                {{3,4,5},{1,2,4,5}}
                                                {{1,5},{2,4},{1,2,3}}
                                                {{2,5},{3,4},{1,2,4}}
                                                {{3,5},{1,2,5},{1,3,4}}
                                                {{4,5},{1,3,5},{2,3,4}}
                                                {{1,4,5},{2,3,5},{1,2,3,4}}

Crossrefs

The non-covering case is A326574.
Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326518, A326534, A326565.

Programs

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

Extensions

a(9) from Andrew Howroyd, Aug 14 2019

A326572 Number of covering antichains of subsets of {1..n}, all having different sums.

Original entry on oeis.org

2, 1, 2, 8, 80, 3015, 803898
Offset: 0

Views

Author

Gus Wiseman, Jul 18 2019

Keywords

Comments

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}.

Examples

			The a(0) = 2 through a(3) = 8 antichains:
  {}    {{1}}  {{1,2}}    {{1,2,3}}
  {{}}         {{1},{2}}  {{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}}
The a(4) = 80 antichains:
  {1234}  {1}{234}    {1}{2}{34}     {1}{2}{3}{4}       {12}{13}{14}{24}{34}
          {12}{34}    {1}{3}{24}     {1}{23}{24}{34}    {12}{13}{23}{24}{34}
          {13}{24}    {1}{4}{23}     {2}{13}{14}{34}
          {2}{134}    {2}{3}{14}     {12}{13}{14}{24}
          {3}{124}    {1}{23}{24}    {12}{13}{14}{34}
          {4}{123}    {1}{23}{34}    {12}{13}{23}{24}
          {12}{134}   {1}{24}{34}    {12}{13}{23}{34}
          {12}{234}   {2}{13}{14}    {12}{13}{24}{34}
          {13}{124}   {2}{13}{34}    {12}{14}{24}{34}
          {13}{234}   {2}{14}{34}    {12}{23}{24}{34}
          {14}{123}   {3}{14}{24}    {13}{14}{24}{34}
          {14}{234}   {4}{12}{23}    {13}{23}{24}{34}
          {23}{124}   {12}{13}{14}   {12}{13}{14}{234}
          {23}{134}   {12}{13}{24}   {12}{23}{24}{134}
          {24}{134}   {12}{13}{34}   {123}{124}{134}{234}
          {34}{123}   {12}{14}{34}
          {123}{124}  {12}{23}{24}
          {123}{134}  {12}{23}{34}
          {123}{234}  {12}{24}{34}
          {124}{134}  {13}{14}{24}
          {124}{234}  {13}{23}{24}
          {134}{234}  {13}{23}{34}
                      {13}{24}{34}
                      {14}{24}{34}
                      {12}{13}{234}
                      {12}{14}{234}
                      {12}{23}{134}
                      {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}

Crossrefs

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 are A326566.
Antichain covers with different edge-sizes are A326570.
The case without singletons is A326571.
Antichains with equal edge-sums are A326574.
Cf. A000372, A035470, A307249, A321469, A326519, A326565, A326569, A326573.

Programs

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

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

Original entry on oeis.org

2, 3, 5, 10, 22, 61, 247, 2096, 81896, 52260575
Offset: 0

Views

Author

Gus Wiseman, Jul 18 2019

Keywords

Comments

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}.

Examples

			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}}

Crossrefs

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.

Programs

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

Extensions

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

Views

Author

Gus Wiseman, Jul 18 2019

Keywords

Comments

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}.

Examples

			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}

Crossrefs

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.

Programs

  • 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]]]];
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

Views

Author

Gus Wiseman, Jul 18 2019

Keywords

Comments

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}.

Examples

			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}

Crossrefs

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.

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

Views

Author

Gus Wiseman, Jul 18 2019

Keywords

Comments

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}.

Examples

			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}}

Crossrefs

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.

Programs

  • 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]]]];
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}]

Formula

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

Extensions

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

Views

Author

Gus Wiseman, Sep 30 2019

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

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

Extensions

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

A336138 Number of set partitions of the binary indices of n with distinct block-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 5, 2, 4, 5, 12, 1, 2, 2, 5, 2, 5, 4, 13, 2, 4, 5, 13, 5, 13, 13, 43, 1, 2, 2, 5, 2, 5, 5, 13, 2, 5, 4, 14, 5, 13, 14, 42, 2, 4, 5, 13, 5, 14, 13, 43, 5, 13, 14, 45, 14, 44, 44, 160, 1, 2, 2, 5, 2, 5, 5, 14, 2, 5, 5, 14, 4, 13
Offset: 0

Views

Author

Gus Wiseman, Jul 12 2020

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Examples

			The a(n) set partitions for n = 3, 7, 11, 15, 23:
  {12}    {123}      {124}      {1234}        {1235}
  {1}{2}  {1}{23}    {1}{24}    {1}{234}      {1}{235}
          {13}{2}    {12}{4}    {12}{34}      {12}{35}
          {1}{2}{3}  {14}{2}    {123}{4}      {123}{5}
                     {1}{2}{4}  {124}{3}      {125}{3}
                                {13}{24}      {13}{25}
                                {134}{2}      {135}{2}
                                {1}{2}{34}    {15}{23}
                                {1}{23}{4}    {1}{2}{35}
                                {1}{24}{3}    {1}{25}{3}
                                {14}{2}{3}    {13}{2}{5}
                                {1}{2}{3}{4}  {15}{2}{3}
                                              {1}{2}{3}{5}

Links

Crossrefs

The version for twice-partitions is A271619.
The version for partitions of partitions is (also) A271619.
These set partitions are counted by A275780.
The version for factorizations is A321469.
The version for normal multiset partitions is A326519.
The version for equal block-sums is A336137.
Set partitions with distinct block-lengths are A007837.
Set partitions of binary indices are A050315.
Twice-partitions with equal sums are A279787.
Partitions of partitions with equal sums are A305551.
Normal multiset partitions with equal block-lengths are A317583.
Multiset partitions with distinct block-sums are ranked by A326535.
Cf. A000110, A032011, A035470, A131632, A321455, A326026, A326514, A326517, A326518, A326534, A326565.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],UnsameQ@@Total/@#&]],{n,0,100}]
Showing 1-8 of 8 results.

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