A319559 -id:A319559 - cp's OEIS frontend

A330109 BII-numbers of BII-normalized set-systems.

0, 1, 3, 4, 5, 7, 11, 12, 13, 15, 20, 21, 22, 23, 30, 31, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 76, 77, 79, 84, 85, 86, 87, 94, 95, 116, 117, 119, 127, 139, 140, 141, 143, 148, 149, 150, 151, 158, 159, 180, 181, 183, 191, 192, 193, 195, 196, 197, 199, 203

Offset: 1

Gus Wiseman, Dec 05 2019

nonn

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 set-system 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.
We define the BII-normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest BII-number.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}

			The sequence of all nonempty BII-normalized set-systems together with their BII-numbers begins:
   1: {1}                  52: {12}{13}{23}
   3: {1}{2}               53: {1}{12}{13}{23}
   4: {12}                 55: {1}{2}{12}{13}{23}
   5: {1}{12}              63: {1}{2}{3}{12}{13}{23}
   7: {1}{2}{12}           64: {123}
  11: {1}{2}{3}            65: {1}{123}
  12: {3}{12}              67: {1}{2}{123}
  13: {1}{3}{12}           68: {12}{123}
  15: {1}{2}{3}{12}        69: {1}{12}{123}
  20: {12}{13}             71: {1}{2}{12}{123}
  21: {1}{12}{13}          75: {1}{2}{3}{123}
  22: {2}{12}{13}          76: {3}{12}{123}
  23: {1}{2}{12}{13}       77: {1}{3}{12}{123}
  30: {2}{3}{12}{13}       79: {1}{2}{3}{12}{123}
  31: {1}{2}{3}{12}{13}    84: {12}{13}{123}

Equals the image/fixed points of the idempotent sequence A330195.
A subset of A326754.
Unlabeled covering set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
BII-weight is A326031.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A330101, A330102.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
biinorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],biinorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],fbi[fbi/@#]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[0,100],Sort[bpe/@bpe[#]]==biinorm[bpe/@bpe[#]]&]

A330110 BII-numbers of lexicographically normalized set-systems.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 13, 15, 20, 21, 23, 31, 33, 37, 45, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 77, 79, 84, 85, 87, 95, 97, 101, 109, 116, 117, 119, 127, 139, 141, 143, 149, 151, 159, 165, 173, 181, 183, 191, 193, 195, 197, 199, 203, 205, 207, 213, 215

Offset: 1

Gus Wiseman, Dec 05 2019

nonn

First differs from A330099 in having 13 and lacking 19.
First differs from A330123 in having 141 and lacking 180, with corresponding set-systems 141: {{1},{3},{4},{1,2}} and 180: {{4},{1,2},{1,3},{2,3}}.
We define the lexicographic normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the lexicographically least of these representatives.
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 set-system 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.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}

			The sequence of all nonempty lexicographically normalized set-systems together with their BII-numbers begins:
   1: {1}                  52: {12}{13}{23}
   3: {1}{2}               53: {1}{12}{13}{23}
   4: {12}                 55: {1}{2}{12}{13}{23}
   5: {1}{12}              63: {1}{2}{3}{12}{13}{23}
   7: {1}{2}{12}           64: {123}
  11: {1}{2}{3}            65: {1}{123}
  13: {1}{3}{12}           67: {1}{2}{123}
  15: {1}{2}{3}{12}        68: {12}{123}
  20: {12}{13}             69: {1}{12}{123}
  21: {1}{12}{13}          71: {1}{2}{12}{123}
  23: {1}{2}{12}{13}       75: {1}{2}{3}{123}
  31: {1}{2}{3}{12}{13}    77: {1}{3}{12}{123}
  33: {1}{23}              79: {1}{2}{3}{12}{123}
  37: {1}{12}{23}          84: {12}{13}{123}
  45: {1}{3}{12}{23}       85: {1}{12}{13}{123}

A subset of A326754.
Unlabeled covering set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
BII-weight is A326031.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A330101, A330102, A330194, A330195.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

A330123 BII-numbers of MM-normalized set-systems.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 13, 15, 20, 21, 23, 31, 33, 37, 45, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 77, 79, 84, 85, 87, 95, 97, 101, 109, 116, 117, 119, 127, 139, 143, 159, 173, 180, 181, 183, 191, 195, 196, 197, 199, 203, 205, 207, 212, 213, 215, 223, 225, 229

Offset: 1

Gus Wiseman, Dec 05 2019

nonn

First differs from A330110 in lacking 141 and having 180, with corresponding set-systems 141: {{1},{3},{4},{1,2}} and 180: {{4},{1,2},{1,3},{2,3}}.
A set-system is a finite set of finite nonempty set of positive integers.
We define the MM-normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest MM-number.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
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 set-system 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.
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all MM-normalized set-systems together with their BII-numbers begins:
   0: {}                           45: {{1},{3},{1,2},{2,3}}
   1: {{1}}                        52: {{1,2},{1,3},{2,3}}
   3: {{1},{2}}                    53: {{1},{1,2},{1,3},{2,3}}
   4: {{1,2}}                      55: {{1},{2},{1,2},{1,3},{2,3}}
   5: {{1},{1,2}}                  63: {{1},{2},{3},{1,2},{1,3},{2,3}}
   7: {{1},{2},{1,2}}              64: {{1,2,3}}
  11: {{1},{2},{3}}                65: {{1},{1,2,3}}
  13: {{1},{3},{1,2}}              67: {{1},{2},{1,2,3}}
  15: {{1},{2},{3},{1,2}}          68: {{1,2},{1,2,3}}
  20: {{1,2},{1,3}}                69: {{1},{1,2},{1,2,3}}
  21: {{1},{1,2},{1,3}}            71: {{1},{2},{1,2},{1,2,3}}
  23: {{1},{2},{1,2},{1,3}}        75: {{1},{2},{3},{1,2,3}}
  31: {{1},{2},{3},{1,2},{1,3}}    77: {{1},{3},{1,2},{1,2,3}}
  33: {{1},{2,3}}                  79: {{1},{2},{3},{1,2},{1,2,3}}
  37: {{1},{1,2},{2,3}}            84: {{1,2},{1,3},{1,2,3}}

A subset of A326754.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
MM-weight is A302242.
Cf. A000120, A000612, A048793, A056239, A070939, A112798, A300913, A319559, A320456, A326031, A330101, A330102, A330194.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[0,100],Sort[bpe/@bpe[#]]==mmnorm[bpe/@bpe[#]]&]

A319560 Number of non-isomorphic strict T_0 multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 6, 15, 40, 121, 353, 1107, 3550, 11818

Offset: 0

Gus Wiseman, Sep 23 2018

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. The T_0 condition means that there are no equivalent vertices.

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(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{1},{2,2}}
   {{2},{1,2}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{1},{2,2,2}}
   {{1},{2,3,3}}
   {{2},{1,2,2}}
   {{1,1},{2,2}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
   {{1},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}

Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A316980.

Cf. A319557, A319558, A319559, A319564, A319565, A319566, A319567.

A326946 Number of unlabeled T_0 set-systems on n vertices.

Original entry on oeis.org

1, 2, 5, 34, 1919, 18660178

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			Non-isomorphic representatives of the a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{1},{2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

The non-T_0 version is A000612.
The antichain case is A245567.
The covering case is A319637.
The labeled version is A326940.
The version with empty edges allowed is A326949.
Cf. A003180, A055621, A059052, A059201, A316978, A319559, A319564, A326942.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Union[normclut/@Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&]]],{n,0,3}]

Partial sums of A319637.

a(n) = A326949(n)/2.

a(5) from Max Alekseyev, Oct 11 2023

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).

A326940 Number of T_0 set-systems on n vertices.

Original entry on oeis.org

1, 2, 7, 112, 32105, 2147161102, 9223372004645756887, 170141183460469231537996491362807709908, 57896044618658097711785492504343953921871039195927143534469727707459805807105

Offset: 0

Gus Wiseman, Aug 07 2019

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(2) = 7 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

The non-T_0 version is A058891 shifted to the left.
The covering case is A059201.
The version with empty edges is A326941.
The unlabeled version is A326946.
Cf. A003180, A316978, A319559, A319564, A319637, A326939, A326947, A326949.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&]],{n,0,3}]

Binomial transform of A059201.

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.

A326941 Number of T_0 sets of subsets of {1..n}.

Original entry on oeis.org

2, 4, 14, 224, 64210, 4294322204, 18446744009291513774, 340282366920938463075992982725615419816, 115792089237316195423570985008687907843742078391854287068939455414919611614210

Offset: 0

Gus Wiseman, Aug 07 2019

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 14 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A001146.
The covering case is A326939.
The case without empty edges is A326940.
The unlabeled version is A326949.
Cf. A003180, A059201, A316978, A319559, A319564, A319637, A326946, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],UnsameQ@@dual[#]&]],{n,0,3}]

a(n) = 2 * A326940(n).

Binomial transform of A326939.

a(5)-a(8) from Andrew Howroyd, Aug 14 2019

A330229 Number of fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 2, 42, 21336

Offset: 0

Gus Wiseman, Dec 08 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.

			The a(3) = 42 set-systems:
  {1}{2}{13}    {1}{2}{12}{13}    {1}{2}{12}{13}{123}
  {1}{2}{23}    {1}{2}{12}{23}    {1}{2}{12}{23}{123}
  {1}{3}{12}    {1}{3}{12}{13}    {1}{3}{12}{13}{123}
  {1}{3}{23}    {1}{3}{13}{23}    {1}{3}{13}{23}{123}
  {2}{3}{12}    {2}{3}{12}{23}    {2}{3}{12}{23}{123}
  {2}{3}{13}    {2}{3}{13}{23}    {2}{3}{13}{23}{123}
  {1}{12}{23}   {1}{2}{13}{123}
  {1}{13}{23}   {1}{2}{23}{123}
  {2}{12}{13}   {1}{3}{12}{123}
  {2}{13}{23}   {1}{3}{23}{123}
  {3}{12}{13}   {2}{3}{12}{123}
  {3}{12}{23}   {2}{3}{13}{123}
  {1}{12}{123}  {1}{12}{23}{123}
  {1}{13}{123}  {1}{13}{23}{123}
  {2}{12}{123}  {2}{12}{13}{123}
  {2}{23}{123}  {2}{13}{23}{123}
  {3}{13}{123}  {3}{12}{13}{123}
  {3}{23}{123}  {3}{12}{23}{123}

The non-covering version is A330282.
Costrict (or T_0) covering set-systems are A059201.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral partitions are counted by A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A055621, A083323, A283877, A319559, A319564, A330224, A330234.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[graprms[#]]==n!&]],{n,0,3}]

Binomial transform is A330282.

cp's OEIS Frontend

A330109 BII-numbers of BII-normalized set-systems.

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A330110 BII-numbers of lexicographically normalized set-systems.

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A330123 BII-numbers of MM-normalized set-systems.

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A319560 Number of non-isomorphic strict T_0 multiset partitions of weight n.

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A326946 Number of unlabeled T_0 set-systems on n vertices.

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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.

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A326940 Number of T_0 set-systems on n vertices.

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

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A326941 Number of T_0 sets of subsets of {1..n}.

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