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A326749 BII-numbers of connected set-systems. 30
0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

The sequence of all connected set-systems together with their BII-numbers begins:

   0: {}

   1: {{1}}

   2: {{2}}

   4: {{1,2}}

   5: {{1},{1,2}}

   6: {{2},{1,2}}

   7: {{1},{2},{1,2}}

   8: {{3}}

  16: {{1,3}}

  17: {{1},{1,3}}

  20: {{1,2},{1,3}}

  21: {{1},{1,2},{1,3}}

  22: {{2},{1,2},{1,3}}

  23: {{1},{2},{1,2},{1,3}}

  24: {{3},{1,3}}

  25: {{1},{3},{1,3}}

  28: {{1,2},{3},{1,3}}

  29: {{1},{1,2},{3},{1,3}}

  30: {{2},{1,2},{3},{1,3}}

  31: {{1},{2},{1,2},{3},{1,3}}

MATHEMATICA

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]]]]]]]]];

Select[Range[0, 100], Length[csm[bpe/@bpe[#]]]<=1&]

CROSSREFS

Positions of 0's and 1's in A326753.

Cf. A000120, A001187, A029931, A048143, A048793, A070939, A072639, A323818, A326031, A326702.

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

Sequence in context: A303393 A039085 A302433 * A327111 A326853 A326879

Adjacent sequences:  A326746 A326747 A326748 * A326750 A326751 A326752

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 23 2019

STATUS

approved

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Last modified April 17 10:38 EDT 2021. Contains 343064 sequences. (Running on oeis4.)