A326751 -id:A326751 - cp's OEIS frontend

A309314 BII-numbers of hyperforests.

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 292, 304, 320, 512, 513, 516, 520, 521, 524, 528, 532

Offset: 1

Gus Wiseman, Jul 23 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. 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. In an antichain, no edge is a subset or superset of any other edge. A hyperforest is an antichain of nonempty sets whose connected components are hypertrees, meaning they have density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices.

			The sequence of all hyperforests together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    4: {{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   12: {{1,2},{3}}
   16: {{1,3}}
   18: {{2},{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   33: {{1},{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}
  130: {{2},{4}}
  131: {{1},{2},{4}}
  132: {{1,2},{4}}
  136: {{3},{4}}
  137: {{1},{3},{4}}

Cf. A000120, A030019, A035053, A048143, A048793, A052888, A070939, A134954, A275307, A326031, A326702, A326753.

Other BII-numbers: A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

A326788 BII-numbers of simple labeled graphs.

Original entry on oeis.org

0, 4, 16, 20, 32, 36, 48, 52, 256, 260, 272, 276, 288, 292, 304, 308, 512, 516, 528, 532, 544, 548, 560, 564, 768, 772, 784, 788, 800, 804, 816, 820, 2048, 2052, 2064, 2068, 2080, 2084, 2096, 2100, 2304, 2308, 2320, 2324, 2336, 2340, 2352, 2356, 2560, 2564

Offset: 1

Gus Wiseman, Jul 25 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.

Also numbers whose binary indices all belong to A018900.

			The sequence of all simple labeled graphs together with their BII-numbers begins:
    0: {}
    4: {{1,2}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
  256: {{1,4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  288: {{2,3},{1,4}}
  292: {{1,2},{2,3},{1,4}}
  304: {{1,3},{2,3},{1,4}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  512: {{2,4}}
  516: {{1,2},{2,4}}
  528: {{1,3},{2,4}}
  532: {{1,2},{1,3},{2,4}}

Cf. A000120, A006125, A006129, A018900, A048793, A062880, A070939, A309356 (same for MM-numbers), A322551, A326031, A326702.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

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

A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

Original entry on oeis.org

4, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 65, 66, 68, 72, 76, 80, 82, 84, 96, 97, 100, 112, 132, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 193, 194, 196, 200, 204, 208, 210, 212, 224, 225, 228, 240, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288

Offset: 1

Gus Wiseman, Dec 11 2023

nonn

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in more than one way.
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. 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 digits (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.
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in only one way (1,2,3), so 21 is not in the sequence.
The terms together with the corresponding set-systems begin:
   4: {{1,2}}
  12: {{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  20: {{1,2},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  36: {{1,2},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Wikipedia, Axiom of choice.

These set-systems are counted by A367772.
Positions of terms > 1 in A367905, firsts A367910, sorted firsts A367911.
If there is at least one choice we get A367906, counted by A367902.
If there are no choices we get A367907, counted by A367903.
If there is one unique choice we get A367908, counted by A367904.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A368098 counts unlabeled multiset partitions per axiom, complement A368097.
Cf. A000612, A055621, A072639, A309326, A326702, A326753, A355529, A368100.
BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers), A326783 (uniform), A326784 (regular), A326788 (simple), A330217 (achiral).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100], Length[Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]]>1&]

A367907 U A367908 U A367909 = A000027.

A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).

Original entry on oeis.org

4, 5, 6, 7, 16, 17, 24, 25, 32, 34, 40, 42, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107

Offset: 1

Gus Wiseman, Aug 22 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 (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.

A set-system is 2-cut-connected if any single vertex can be removed (along with any empty edges) without making the set-system disconnected or empty. Except for cointersecting set-systems (A326853), this is the same as 2-vertex-connectivity.

			The sequence of all 2-cut-connected set-systems together with their BII-numbers begins:
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  63: {{1},{2},{1,2},{3},{1,3},{2,3}}

Positions of numbers >= 2 in A326786.
2-cut-connected graphs are counted by A013922, if we assume A013922(2) = 0.
2-cut-connected integer partitions are counted by A322387.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for cut-connectivity 1 are A327098.
BII-numbers for non-spanning edge-connectivity >= 2 are A327102.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Covering 2-cut-connected set-systems are counted by A327112.
Covering set-systems with cut-connectivity 2 are counted by A327113.
The labeled cut-connectivity triangle is A327125, with unlabeled version A327127.
Cf. A000120, A002218, A048793, A070939, A259862, A322388, A326031, A326749, A327097, A327108.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
Select[Range[0,100],cutConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]>=2&]

If (*) is intersection and (-) is complement, we have A327101 * A326704 = A326751 - A058891, i.e., the intersection of A327101 (this sequence) with A326704 (antichains) is the complement of A058891 (singletons) in A326751 (blobs).

A327100 BII-numbers of antichains of sets with cut-connectivity 1.

Original entry on oeis.org

1, 2, 8, 20, 36, 48, 128, 260, 272, 276, 292, 304, 308, 320, 516, 532, 544, 548, 560, 564, 576, 768, 784, 788, 800, 804, 1040, 1056, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304, 2308, 2324, 2336, 2352, 2560, 2564, 2576, 2596, 2608, 2816, 2820, 2832, 2848

Offset: 1

Gus Wiseman, Aug 22 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 (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.

We define the cut-connectivity of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).

			The sequence of all antichains of sets with vertex-connectivity 1 together with their BII-numbers begins:
    1: {{1}}
    2: {{2}}
    8: {{3}}
   20: {{1,2},{1,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
  128: {{4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  292: {{1,2},{2,3},{1,4}}
  304: {{1,3},{2,3},{1,4}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  320: {{1,2,3},{1,4}}
  516: {{1,2},{2,4}}
  532: {{1,2},{1,3},{2,4}}
  544: {{2,3},{2,4}}
  548: {{1,2},{2,3},{2,4}}
  560: {{1,3},{2,3},{2,4}}
  564: {{1,2},{1,3},{2,3},{2,4}}

Positions of 1's in A326786.
The graphical case is A327114.
BII numbers of antichains with vertex-connectivity >= 1 are A326750.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for cut-connectivity 1 are A327098.
Cf. A000120, A000372, A006126, A048143, A048793, A070939, A322390, A326031, A326749, A326751, A327071, A327111.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
Select[Range[0,100],stableQ[bpe/@bpe[#],SubsetQ]&&cutConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]

If (+) is union and (-) is complement, we have A327100 = A058891 + (A326750 - A326751).

cp's OEIS Frontend

A309314 BII-numbers of hyperforests.

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A326788 BII-numbers of simple labeled graphs.

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A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

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A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).

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A327100 BII-numbers of antichains of sets with cut-connectivity 1.

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