A326875 -id:A326875 - cp's OEIS frontend

A326873 BII-numbers of connectedness systems without singletons.

0, 4, 16, 32, 64, 68, 80, 84, 96, 100, 112, 116, 256, 288, 512, 528, 1024, 1028, 1280, 1284, 1536, 1540, 1792, 1796, 2048, 2052, 4096, 4112, 4352, 4368, 6144, 6160, 6400, 6416, 8192, 8224, 8704, 8736, 10240, 10272, 10752, 10784, 16384, 16388, 16400, 16416

Offset: 1

Gus Wiseman, Jul 29 2019

nonn

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.
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 enumeration of these set-systems by number of covered vertices is given by A326877.

			The sequence of all connectedness systems without singletons together with their BII-numbers begins:
     0: {}
     4: {{1,2}}
    16: {{1,3}}
    32: {{2,3}}
    64: {{1,2,3}}
    68: {{1,2},{1,2,3}}
    80: {{1,3},{1,2,3}}
    84: {{1,2},{1,3},{1,2,3}}
    96: {{2,3},{1,2,3}}
   100: {{1,2},{2,3},{1,2,3}}
   112: {{1,3},{2,3},{1,2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   256: {{1,4}}
   288: {{2,3},{1,4}}
   512: {{2,4}}
   528: {{1,3},{2,4}}
  1024: {{1,2,4}}
  1028: {{1,2},{1,2,4}}
  1280: {{1,4},{1,2,4}}
  1284: {{1,2},{1,4},{1,2,4}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Connectedness systems without singletons are counted by A072446, with unlabeled case A072444.
Connectedness systems are counted by A326866, with unlabeled case A326867.
BII-numbers of connectedness systems are A326872.
The connected case is A326879.
Cf. A029931, A048793, A072447, A326031, A326749, A326750, A326870, A326875, A326877.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
connnosQ[eds_]:=!MemberQ[Length/@eds,1]&&SubsetQ[eds,Union@@@Select[Tuples[eds,2],Intersection@@#!={}&]];
Select[Range[0,1000],connnosQ[bpe/@bpe[#]]&]

A326905 BII-numbers of set-systems (without {}) closed under intersection.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 21, 24, 32, 34, 38, 40, 56, 64, 65, 66, 68, 69, 70, 72, 80, 81, 85, 88, 96, 98, 102, 104, 120, 128, 256, 257, 261, 273, 277, 321, 325, 337, 341, 384, 512, 514, 518, 546, 550, 578, 582, 610, 614, 640, 896, 1024, 1025, 1026, 1028

Offset: 1

Gus Wiseman, Aug 04 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 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 closed under intersection together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  21: {{1},{1,2},{1,3}}
  24: {{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  38: {{2},{1,2},{2,3}}
  40: {{3},{2,3}}
  56: {{3},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}

The case with union instead of intersection is A326875.
The case closed under union and intersection is A326913.
Set-systems closed under intersection and containing the vertex set are A326903.
Set-systems closed under intersection are A326901, with unlabeled version A326904.
Cf. A006058, A102895, A102898, A326866, A326876, A326878, A326882, A326900, A326902.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Intersection@@@Tuples[bpe/@bpe[#],2]]&]

A326913 BII-numbers of set-systems (without {}) closed under union and intersection.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 24, 32, 34, 40, 64, 65, 66, 68, 69, 70, 72, 80, 81, 85, 88, 96, 98, 102, 104, 120, 128, 256, 257, 384, 512, 514, 640, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1285, 1408, 1536, 1538, 1542, 1664, 1920, 2048, 2056, 2176

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

			The sequence of all set-systems closed under union and intersection together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Cf. A048793, A102894, A102895, A102896, A102897, A326031, A326875, A326876, A326878, A326880, A326901.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union@@@Tuples[bpe/@bpe[#],2]]&&SubsetQ[bpe/@bpe[#],Intersection@@@Tuples[bpe/@bpe[#],2]]&]

A327016 BII-numbers of finite T_0 topologies without their empty set.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 8, 17, 24, 25, 34, 40, 42, 69, 70, 71, 81, 85, 87, 88, 89, 93, 98, 102, 103, 104, 106, 110, 120, 121, 122, 127, 128, 257, 384, 385, 514, 640, 642, 1029, 1030, 1031, 1281, 1285, 1287, 1408, 1409, 1413, 1538, 1542, 1543, 1664, 1666, 1670, 1920

Offset: 1

Gus Wiseman, Aug 14 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}}. The T_0 condition means that the dual is strict (no repeated edges).

			The sequence of all finite T_0 topologies without their empty set together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
  81: {{1},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  87: {{1},{2},{1,2},{1,3},{1,2,3}}
  88: {{3},{1,3},{1,2,3}}

Gus Wiseman, The first 100 finite T_0 topologies, represented as posets.

T_0 topologies are A001035, with unlabeled version A000112.
BII-numbers of topologies without their empty set are A326876.
BII-numbers of T_0 set-systems are A326947.
Cf. A001930, A048793, A306445, A316978, A319564, A326031, A326872, A326875, A326939, A326941, A326959.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],UnsameQ@@dual[bpe/@bpe[#]]&&SubsetQ[bpe/@bpe[#],Union[Union@@@Tuples[bpe/@bpe[#],2],DeleteCases[Intersection@@@Tuples[bpe/@bpe[#],2],{}]]]&]

cp's OEIS Frontend

A326873 BII-numbers of connectedness systems without singletons.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A326905 BII-numbers of set-systems (without {}) closed under intersection.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A326913 BII-numbers of set-systems (without {}) closed under union and intersection.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327016 BII-numbers of finite T_0 topologies without their empty set.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica