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A309326 BII-numbers of minimal covers. 0
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, 320, 512, 513, 516, 520, 521, 524, 528, 544, 545, 548 (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. 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. A minimal cover is a set-system where every edge contains at least one vertex that does not belong to any other edge.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

The sequence of all minimal covers 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}}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Select[Range[0, 1000], And@@Table[Union@@Delete[bpe/@bpe[#], i]!=Union@@bpe/@bpe[#], {i, Length[bpe/@bpe[#]]}]&]

CROSSREFS

Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.

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

Sequence in context: A165564 A326704 A309314 * A326701 A061887 A005455

Adjacent sequences:  A309323 A309324 A309325 * A309327 A309328 A309329

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 23 2019

STATUS

approved

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Last modified May 21 08:44 EDT 2022. Contains 353908 sequences. (Running on oeis4.)