A326566 -id:A326566 - cp's OEIS frontend

A326565 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.

1, 0, 1, 1, 4, 13, 91, 1318, 73581, 51913025

Offset: 0

Gus Wiseman, Jul 13 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(2) = 1 through a(5) = 13 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}      {{1,2,3,4,5}}
                      {{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}}
                                       {{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}}

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326360, A326518, A326534, A326566.

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}]

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

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(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}

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.

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

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.

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

A327901 Nonprime squarefree numbers whose prime indices all have the same sum of prime indices (A056239).

Original entry on oeis.org

1, 35, 143, 209, 247, 391, 493, 629, 667, 851, 901, 1073, 1219, 1333, 1457, 1537, 1891, 1961, 2021, 2201, 2623, 2717, 2759, 2867, 2993, 3053, 3239, 3337, 3827, 3977, 4061, 4183, 4223, 4331, 4387, 4633, 5429, 5633, 5767, 5959, 6157, 6191, 6319, 7081, 7093, 7519

Offset: 1

Gus Wiseman, Sep 30 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
     1: {}
    35: {3,4}
   143: {5,6}
   209: {5,8}
   247: {6,8}
   391: {7,9}
   493: {7,10}
   629: {7,12}
   667: {9,10}
   851: {9,12}
   901: {7,16}
  1073: {10,12}
  1219: {9,16}
  1333: {11,14}
  1457: {11,15}
  1537: {10,16}
  1891: {11,18}
  1961: {12,16}
  2021: {14,15}
  2201: {11,20}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version including primes and nonsquarefree numbers is A326534.
The version for number of prime indices is A327900.
The version for mean of prime indices is A327902.
Cf. A001222, A005117, A056239, A112798, A302242, A306021, A320324, A326566, A326574, A327908.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@Total/@primeMS/@primeMS[#]&];

cp's OEIS Frontend

A326565 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.

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

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

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A327901 Nonprime squarefree numbers whose prime indices all have the same sum of prime indices (A056239).

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