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A326873 BII-numbers of connectedness systems without singletons. 5
0, 4, 16, 32, 64, 68, 80, 84, 96, 100, 112, 116, 256, 288, 512, 528, 1024, 1028, 1280, 1284, 1536, 1540, 1792, 1796, 2048, 2052, 4096, 4112, 4352, 4368, 6144, 6160, 6400, 6416, 8192, 8224, 8704, 8736, 10240, 10272, 10752, 10784, 16384, 16388, 16400, 16416 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.

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.

The enumeration of these set-systems by number of covered vertices is given by A326877.

LINKS

Table of n, a(n) for n=1..46.

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

EXAMPLE

The sequence of all connectedness systems without singletons together with their BII-numbers begins:

     0: {}

     4: {{1,2}}

    16: {{1,3}}

    32: {{2,3}}

    64: {{1,2,3}}

    68: {{1,2},{1,2,3}}

    80: {{1,3},{1,2,3}}

    84: {{1,2},{1,3},{1,2,3}}

    96: {{2,3},{1,2,3}}

   100: {{1,2},{2,3},{1,2,3}}

   112: {{1,3},{2,3},{1,2,3}}

   116: {{1,2},{1,3},{2,3},{1,2,3}}

   256: {{1,4}}

   288: {{2,3},{1,4}}

   512: {{2,4}}

   528: {{1,3},{2,4}}

  1024: {{1,2,4}}

  1028: {{1,2},{1,2,4}}

  1280: {{1,4},{1,2,4}}

  1284: {{1,2},{1,4},{1,2,4}}

MATHEMATICA

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

connnosQ[eds_]:=!MemberQ[Length/@eds, 1]&&SubsetQ[eds, Union@@@Select[Tuples[eds, 2], Intersection@@#!={}&]];

Select[Range[0, 1000], connnosQ[bpe/@bpe[#]]&]

CROSSREFS

Connectedness systems without singletons are counted by A072446, with unlabeled case A072444.

Connectedness systems are counted by A326866, with unlabeled case A326867.

BII-numbers of connectedness systems are A326872.

The connected case is A326879.

Cf. A029931, A048793, A072447, A326031, A326749, A326750, A326870, A326875, A326877.

Sequence in context: A031050 A243980 A119677 * A126032 A296819 A034713

Adjacent sequences:  A326870 A326871 A326872 * A326874 A326875 A326876

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 29 2019

STATUS

approved

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Last modified May 16 20:11 EDT 2022. Contains 353720 sequences. (Running on oeis4.)