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A327081 BII-numbers of maximal uniform set-systems covering an initial interval of positive integers. 3
1, 3, 4, 11, 52, 64, 139, 2868, 13376, 16384, 32907 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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..11.

EXAMPLE

The sequence of all maximal uniform set-systems covering an initial interval together with their BII-numbers begins:

      0: {}

      1: {{1}}

      3: {{1},{2}}

      4: {{1,2}}

     11: {{1},{2},{3}}

     52: {{1,2},{1,3},{2,3}}

     64: {{1,2,3}}

    139: {{1},{2},{3},{4}}

   2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

  16384: {{1,2,3,4}}

  32907: {{1},{2},{3},{4},{5}}

MATHEMATICA

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

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

Select[Range[1000], With[{sys=bpe/@bpe[#]}, #==0||normQ[Union@@sys]&&SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys], Length[First[sys]]]]&]

CROSSREFS

BII-numbers of uniform set-systems are A326783.

BII-numbers of maximal uniform set-systems are A327080.

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

Sequence in context: A296256 A101982 A041947 * A201970 A102013 A192223

Adjacent sequences:  A327078 A327079 A327080 * A327082 A327083 A327084

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 20 2019

STATUS

approved

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Last modified July 9 22:46 EDT 2020. Contains 335570 sequences. (Running on oeis4.)