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A327080 BII-numbers of maximal uniform set-systems (or complete hypergraphs). 3
0, 1, 2, 3, 4, 8, 9, 10, 11, 16, 32, 52, 64, 128, 129, 130, 131, 136, 137, 138, 139, 256, 512, 772, 1024, 2048, 2320, 2592, 2868, 4096, 8192, 13376, 16384, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906 (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 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 uniform if all edges have the same size.

LINKS

Table of n, a(n) for n=1..48.

EXAMPLE

The sequence of all maximal uniform set-systems 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}}

   16: {{1,3}}

   32: {{2,3}}

   52: {{1,2},{1,3},{2,3}}

   64: {{1,2,3}}

  128: {{4}}

  129: {{1},{4}}

  130: {{2},{4}}

  131: {{1},{2},{4}}

  136: {{3},{4}}

  137: {{1},{3},{4}}

  138: {{2},{3},{4}}

MATHEMATICA

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

Select[Range[0, 100], With[{sys=bpe/@bpe[#]}, #==0||SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys], Length[First[sys]]]]&]

CROSSREFS

BII-numbers of uniform set-systems are A326783.

The normal case (where the edges cover an initial interval) is A327081.

Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.

Sequence in context: A004826 A326783 A326785 * A291621 A329268 A288860

Adjacent sequences:  A327077 A327078 A327079 * A327081 A327082 A327083

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 20 2019

STATUS

approved

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Last modified July 8 03:38 EDT 2020. Contains 335504 sequences. (Running on oeis4.)