cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A329560 BII-numbers of antichains of sets with empty intersection.

Original entry on oeis.org

0, 3, 9, 10, 11, 12, 18, 33, 52, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 258, 264, 266, 268, 274, 288, 292, 304, 308, 513, 520, 521, 524, 528, 532, 545, 560, 564, 772, 776, 780, 784, 788, 800, 804, 816, 820, 832
Offset: 1

Views

Author

Gus Wiseman, Nov 28 2019

Keywords

Comments

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 of positive integers) 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 an antichain if no edge is a proper subset of any other.
Empty intersection means there is no vertex in common to all the edges

Examples

			The sequence of terms together with their binary expansions and corresponding set-systems begins:
    0:          0 ~ {}
    3:         11 ~ {{1},{2}}
    9:       1001 ~ {{1},{3}}
   10:       1010 ~ {{2},{3}}
   11:       1011 ~ {{1},{2},{3}}
   12:       1100 ~ {{1,2},{3}}
   18:      10010 ~ {{2},{1,3}}
   33:     100001 ~ {{1},{2,3}}
   52:     110100 ~ {{1,2},{1,3},{2,3}}
  129:   10000001 ~ {{1},{4}}
  130:   10000010 ~ {{2},{4}}
  131:   10000011 ~ {{1},{2},{4}}
  132:   10000100 ~ {{1,2},{4}}
  136:   10001000 ~ {{3},{4}}
  137:   10001001 ~ {{1},{3},{4}}
  138:   10001010 ~ {{2},{3},{4}}
  139:   10001011 ~ {{2},{3},{4}}
  140:   10001100 ~ {{1,2},{3},{4}}
  144:   10010000 ~ {{1,3},{4}}
  146:   10010010 ~ {{2},{1,3},{4}}
  148:   10010100 ~ {{1,2},{1,3},{4}}

Crossrefs

Intersection of A326911 and A326704.
BII-numbers of intersecting set-systems with empty intersecting are A326912.
Cf. A000120, A048793, A070939, A326031, A326701, A328671, A329561, A329626, A329628, A329661.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],#==0||Intersection@@bpe/@bpe[#]=={}&&stableQ[bpe/@bpe[#],SubsetQ]&]

A329561 BII-numbers of intersecting antichains of sets.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 52, 64, 128, 256, 260, 272, 276, 320, 512, 516, 544, 548, 576, 768, 772, 832, 1024, 1040, 1056, 1072, 1088, 2048, 2064, 2080, 2096, 2112, 2304, 2320, 2368, 2560, 2592, 2624, 2816, 2880, 3072, 3088, 3104, 3120, 3136, 4096
Offset: 1

Views

Author

Gus Wiseman, Nov 28 2019

Keywords

Comments

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 of positive integers) 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 intersecting if no two edges are disjoint. It is an antichain if no edge is a proper subset of any other.

Examples

			The sequence of terms together with their corresponding set-systems begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    8: {{3}}
   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}}
   64: {{1,2,3}}
  128: {{4}}
  256: {{1,4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  320: {{1,2,3},{1,4}}
  512: {{2,4}}
  516: {{1,2},{2,4}}

Crossrefs

Intersection of A326704 (antichains) and A326910 (intersecting).
Covering intersecting antichains of sets are counted by A305844.
BII-numbers of antichains with empty intersection are A329560.
Cf. A000120, A048143, A048793, A070939, A087086, A305857, A306007, A326031, A326361, A326912, A329628.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ[#1,#2]||Intersection[#1,#2]=={}&]&]

A329626 Smallest BII-number of an antichain with n edges.

Original entry on oeis.org

0, 1, 3, 11, 139, 820, 2868, 35636, 199476, 723764
Offset: 0

Views

Author

Gus Wiseman, Nov 28 2019

Keywords

Comments

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 of positive integers) 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 an antichain if no edge is a proper subset of any other.

Examples

			The sequence of terms together with their corresponding set-systems begins:
       0: {}
       1: {{1}}
       3: {{1},{2}}
      11: {{1},{2},{3}}
     139: {{1},{2},{3},{4}}
     820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
    2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
   35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
  199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
  723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}

Crossrefs

The connected case is A329627.
The intersecting case is A329628.
BII-numbers of antichains are A326704.
Antichain covers are A006126.
Cf. A048143, A048793, A070939, A303362, A319837, A326031, A326750, A329555, A329560, A329561, A329625.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_]:=!Apply[Or,Outer[#1=!=#2&&SubsetQ[#1,#2]&,u,u,1],{0,1}];
First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&],Length[bpe[#]]&]

A329627 Smallest BII-number of a clutter (connected antichain) with n edges.

Original entry on oeis.org

0, 1, 20, 52, 308, 820, 2868, 68404, 199476, 723764
Offset: 0

Views

Author

Gus Wiseman, Nov 28 2019

Keywords

Comments

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 of positive integers) 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 an antichain if no edge is a proper subset of any other.
For n > 1, a(n) appears to be the number whose binary indices are the first n terms of A018900.

Examples

			The sequence of terms together with their corresponding set-systems begins:
       0: {}
       1: {{1}}
      20: {{1,2},{1,3}}
      52: {{1,2},{1,3},{2,3}}
     308: {{1,2},{1,3},{2,3},{1,4}}
     820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
    2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
   68404: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5}}
  199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
  723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}

Crossrefs

The version for MM-numbers is A329555.
BII-numbers of clutters are A326750.
Clutters of sets are counted by A048143.
Minimum BII-numbers of connected set-systems are A329625.
Minimum BII-numbers of antichains are A329626.
MM-numbers of connected weak antichains of multisets are A329559.
Cf. A048793, A070939, A072639, A320275, A322113, A326031, A326704, A326753, A329628, A329632.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&&Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]
Showing 1-4 of 4 results.

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