A326971 -id:A326971 - cp's OEIS frontend

A326965 Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.

1, 2, 5, 46, 19181, 2010327182, 9219217424630040409, 170141181796805106025395618012972506978, 57896044618658097536026644159052312978532934306727333157337631572314050272137

Offset: 0

Gus Wiseman, Aug 10 2019

nonn

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set-system where no edge is a subset of any other. This sequence counts set-systems whose dual is a (strict) antichain, also called T_1 set-systems.

			The a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{2},{1,2}}

Set-systems are A058891.
T_0 set-systems are A326940.
The covering case is A326961.
The version with empty edges allowed is A326967.
Set-systems whose dual is a weak antichain are A326968.
The unlabeled version is A326972.
The BII_numbers of these set-systems are A326979.
Cf. A059052, A326951, A326966, A326970, A326971, A326976, A326977.

tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],tmQ]],{n,0,3}]

Binomial transform of A326961.

a(n) = A326967(n)/2.

A326974 Number of unlabeled set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 set-systems.

Original entry on oeis.org

1, 1, 2, 16, 1212

Offset: 0

Gus Wiseman, Aug 11 2019

Alternatively, these are unlabeled set-systems covering n vertices whose dual is a (strict) antichain. A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set-system where no edge is a subset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 16 set-systems:
  {}  {{1}}  {{1},{2}}        {{1},{2},{3}}
             {{1},{2},{1,2}}  {{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Unlabeled covers are A055621.
The same with T_0 instead of T_1 is A319637.
The labeled version is A326961.
The non-covering version is A326972 (partial sums).
Unlabeled covering set-systems whose dual is a weak antichain are A326973.
Cf. A000612, A059523, A319559, A326946, A326951, A326965, A326970, A326971, A326976, A326977, A326979.

a(n > 0) = A326972(n) - A326972(n - 1).

A326970 Number of set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 3, 43, 19251

Offset: 0

Gus Wiseman, Aug 10 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edges consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(3) = 43 set-systems:
  {123}  {1}{23}  {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}  {12}{13}{23}  {1}{2}{3}{13}
         {3}{12}  {1}{23}{123}  {1}{2}{3}{23}
                  {2}{13}{123}  {1}{2}{13}{23}
                  {3}{12}{123}  {1}{2}{3}{123}
                                {1}{3}{12}{23}
                                {2}{3}{12}{13}
                                {1}{12}{13}{23}
                                {2}{12}{13}{23}
                                {3}{12}{13}{23}
                                {12}{13}{23}{123}
.
  {1}{2}{3}{12}{13}     {1}{2}{3}{12}{13}{23}    {1}{2}{3}{12}{13}{23}{123}
  {1}{2}{3}{12}{23}     {1}{2}{3}{12}{13}{123}
  {1}{2}{3}{13}{23}     {1}{2}{3}{12}{23}{123}
  {1}{2}{12}{13}{23}    {1}{2}{3}{13}{23}{123}
  {1}{2}{3}{12}{123}    {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{13}{123}    {1}{3}{12}{13}{23}{123}
  {1}{2}{3}{23}{123}    {2}{3}{12}{13}{23}{123}
  {1}{3}{12}{13}{23}
  {2}{3}{12}{13}{23}
  {1}{2}{13}{23}{123}
  {1}{3}{12}{23}{123}
  {2}{3}{12}{13}{123}
  {1}{12}{13}{23}{123}
  {2}{12}{13}{23}{123}
  {3}{12}{13}{23}{123}

Covering set-systems are A003465.
Covering set-systems whose dual is strict are A059201.
The T_1 case is A326961.
The BII-numbers of these set-systems are A326966.
The non-covering case is A326968.
The unlabeled version is A326973.
Cf. A006126, A059052, A059523, A326950, A326960, A326965, A326969, A326971, A326974, A326975, A326978.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Inverse binomial transform of A326968.

A326972 Number of unlabeled set-systems on n vertices whose dual is a (strict) antichain, also called unlabeled T_1 set-systems.

Original entry on oeis.org

1, 2, 4, 20, 1232

Offset: 0

Gus Wiseman, Aug 11 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of sets, none of which is a subset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 20 set-systems:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1},{2}}        {{1},{2}}
             {{1},{2},{1,2}}  {{1},{2},{3}}
                              {{1},{2},{1,2}}
                              {{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Unlabeled set-systems are A000612.
Unlabeled set-systems whose dual is strict are A326946.
The version with empty edges allowed is A326951.
The labeled version is A326965.
The version where the dual is not required to be strict is A326971.
The covering version is A326974 (first differences).
Cf. A059523, A319559, A319637, A326973, A326976, A326977, A326979.

A326966 BII-numbers of set-systems whose dual is a weak antichain.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 15, 16, 18, 25, 27, 30, 31, 32, 33, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 64, 75, 76, 79, 82, 91, 94, 95, 97, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 135

Offset: 1

Gus Wiseman, Aug 13 2019

nonn

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

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. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all set-systems whose dual is a weak antichain together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  25: {{1},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}

Set-systems whose dual is a weak antichain are counted by A326968, with covering case A326970, unlabeled version A326971, and unlabeled covering version A326973.
BII-numbers of set-systems whose dual is strict (T_0) are A326947.
BII-numbers of set-systems whose dual is a (strict) antichain (T_1) are A326979.
Cf. A059523, A319639, A326961, A326965, A326969, A326974, A326975, A326978.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[dual[bpe/@bpe[#]],SubsetQ]&]

A326973 Number of unlabeled set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 3, 19, 1243

Offset: 0

Gus Wiseman, Aug 11 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 19 set-systems:
  {}  {{1}}  {{1,2}}          {{1,2,3}}
             {{1},{2}}        {{1},{2,3}}
             {{1},{2},{1,2}}  {{1},{2},{3}}
                              {{1,2},{1,3},{2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Unlabeled covering set-systems are A055621.
The labeled version is A326970.
The non-covering case is A326971 (partial sums).
The case that is also T_0 is the T_1 case A326974.
Cf. A000612, A059523, A319637, A326966, A326968, A326972, A326975, A326978.

A326968 Number of set-systems on n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 2, 6, 56, 19446

Offset: 0

Gus Wiseman, Aug 10 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{2},{1,2}}

The case with strict dual is A326965.
The BII-numbers of these set-systems are A326966.
The version with empty edges allowed is A326969.
The covering case is A326970.
The unlabeled version is A326971.
Cf. A058891, A059523, A326940, A326972, A326973, A326975, A326978, A326979.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

a(n) = A326969(n)/2.

Binomial transform of A326970.

A326975 Number of factorizations of n into factors > 1 whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 3, 2, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 1, 5, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 11, 2, 5, 1, 2, 2, 5, 1, 12, 1, 2, 2, 2, 2, 5, 1, 5, 5, 2, 1, 4, 2, 2

Offset: 1

Gus Wiseman, Aug 13 2019

nonn

The dual of a multiset system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The dual of a factorization is the dual of the multiset partition obtained by replacing each factor with its multiset of prime indices.

A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other.

			The a(36) = 9 factorizations:
  (36)
  (4*9)
  (6*6)
  (2*18)
  (3*12)
  (2*2*9)
  (2*3*6)
  (3*3*4)
  (2*2*3*3)

The T_0 case (where the dual is strict) is A316978.
Set-systems whose dual is a weak antichain are A326968.
Partitions whose dual is a weak antichain are A326978.
The T_1 case (where the dual is a strict antichain) is A327012.
Cf. A001055, A059523, A316978, A319639, A326965, A326966, A326969, A326970, A326971, A326976, A326977.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[facs[n],stableQ[dual[primeMS/@#],submultQ]&]],{n,100}]

A326978 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a weak antichain.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 52, 68, 91, 116, 149, 191, 249, 311, 399, 498, 622, 773, 971, 1193, 1478, 1811, 2222, 2709, 3311, 4021, 4882, 5894, 7110, 8554, 10273, 12312, 14734, 17578, 20941, 24905, 29570, 35056, 41475, 48983, 57752, 68025, 79988

Offset: 0

Gus Wiseman, Aug 13 2019

nonn

The dual of a multiset partition has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}.

A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other.

			The a(0) = 1 through a(7) = 15 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (321)     (331)
                                (11111)  (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (3211)
                                         (111111)  (4111)
                                                   (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)

Set-systems whose dual is a weak antichain are A326968.
Factorizations whose dual is a weak antichain are A326975.
The version where the dual is a strict antichain is A326977.
Cf. A000041, A319564, A319728, A326966, A326969, A326970, A326971, A326973.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
Table[Length[Select[IntegerPartitions[n],stableQ[dual[primeMS/@#],submultQ]&]],{n,0,30}]

A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

2, 4, 12, 112, 38892

Offset: 0

Gus Wiseman, Aug 10 2019

The dual of a set of subsets has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(0) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{1,2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{},{1,2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Sets of subsets whose dual is strict are A326941.
The BII-numbers of set-systems whose dual is a weak antichain are A326966.
Sets of subsets whose dual is a (strict) antichain are A326967.
The case without empty edges is A326968.
Cf. A001146, A059052, A326951, A326970, A326971, A326975, A326978.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n]]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

a(n) = 2 * A326968(n).

a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).

cp's OEIS Frontend

A326965 Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A326974 Number of unlabeled set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 set-systems.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A326970 Number of set-systems covering n vertices whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A326972 Number of unlabeled set-systems on n vertices whose dual is a (strict) antichain, also called unlabeled T_1 set-systems.

Views

Author

Keywords

Comments

Examples

Crossrefs

A326966 BII-numbers of set-systems whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A326973 Number of unlabeled set-systems covering n vertices whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

A326968 Number of set-systems on n vertices whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A326975 Number of factorizations of n into factors > 1 whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A326978 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica