A304986 -id:A304986 - cp's OEIS frontend

A326750 BII-numbers of clutters (connected antichains of nonempty sets).

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 52, 64, 128, 256, 260, 272, 276, 292, 304, 308, 320, 512, 516, 532, 544, 548, 560, 564, 576, 768, 772, 784, 788, 800, 804, 816, 820, 832, 1024, 1040, 1056, 1072, 1088, 2048, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304

Offset: 1

Gus Wiseman, Jul 23 2019

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.

			The sequence of all clutters together with their BII-numbers 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}}
  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}}

John Tyler Rascoe, Table of n, a(n) for n = 1..6834
John Tyler Rascoe, Python program.

The number of clutters spanning n vertices is A048143(n).
Cf. A000120, A001187, A048793, A070939, A072639, A304986, A326031, A326702, A326753.
Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

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}];
Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&]

Python
```
# see linked program
```

Intersection of A326749 and A326704.

A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 40, 1344, 203136, 495598592, 309065330371840, 14369391920653644779049472

Offset: 0

Gus Wiseman, May 23 2018

nonn

Only the non-singleton edges are required to form an antichain.

			The a(2) = 4 clutters:
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Cf. A000372, A006126, A014466, A304997, A304999, A306505, A317674, A319721.

For n > 1, a(n) = A048143(n) * 2^n.

A304982 Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.

Original entry on oeis.org

1, 2, 5, 19, 137, 3053, 822526

Offset: 0

Gus Wiseman, May 23 2018

The initial terms 1, 2, 5, 19 are the same as A304981 but the remaining terms differ.

			Non-isomorphic representatives of the a(3) = 19 clutters:
{}
{{1}}
{{1,2}}
{{1,2,3}}
{{2},{1,2}}
{{1,3},{2,3}}
{{3},{1,2,3}}
{{1},{2},{1,2}}
{{1,2},{1,3},{2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{2},{3},{1,2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{3},{1,2},{1,3},{2,3}}
{{2},{3},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Partial sums of A304983.

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 3, 14, 118, 2916, 819473

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 14 clutters:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986, A304997.

Inverse Euler transform of A304997. - Andrew Howroyd, Aug 14 2019

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304981 Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.

Original entry on oeis.org

1, 1, 2, 5, 19, 176, 16118, 489996568

Offset: 0

Gus Wiseman, May 23 2018

nonn

			Non-isomorphic representatives of the a(3) = 5 clutters:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}
Non-isomorphic representatives of the a(4) = 19 clutters:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,2,3,4}}
  {{1,3},{2,3}}
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Partial sums of A261006(n > 0).

A304984 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.

Original entry on oeis.org

1, 2, 7, 56, 1533, 210302, 496838435, 309068803876372, 14369391923126181310256825

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(2) = 7 clutters:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Binomial transform of A304985(n > 0).

A305005 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} without singleton edges.

Original entry on oeis.org

1, 1, 2, 9, 111, 6829, 7783192, 2414627236071, 56130437209370100252463

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(3) = 9 clutters:
  {}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Binomial transform of A048143 if we assume A048143(1) = 0.

a(n) = A198085(n) - n + 1. - Gus Wiseman, Jun 11 2018

cp's OEIS Frontend

A326750 BII-numbers of clutters (connected antichains of nonempty sets).

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A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

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A304982 Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.

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A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Views

Author

Keywords

Examples

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Formula

Extensions

A304981 Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.

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Keywords

Examples

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A304984 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.

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Author

Keywords

Examples

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Formula

A305005 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} without singleton edges.

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