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A326947 BII-numbers of T_0 set-systems. 21
0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 70, 71, 73, 74, 75, 77, 78 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The dual of a set-system has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated 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.

LINKS

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

EXAMPLE

The sequence of all T_0 set-systems together with their BII numbers begins:

   0: {}

   1: {{1}}

   2: {{2}}

   3: {{1},{2}}

   5: {{1},{1,2}}

   6: {{2},{1,2}}

   7: {{1},{2},{1,2}}

   8: {{3}}

   9: {{1},{3}}

  10: {{2},{3}}

  11: {{1},{2},{3}}

  13: {{1},{1,2},{3}}

  14: {{2},{1,2},{3}}

  15: {{1},{2},{1,2},{3}}

  17: {{1},{1,3}}

  19: {{1},{2},{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}}

MATHEMATICA

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

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];

TZQ[sys_]:=UnsameQ@@dual[sys];

Select[Range[0, 100], TZQ[bpe/@bpe[#]]&]

CROSSREFS

T_0 set-systems are counted by A326940, with unlabeled version A326946.

Cf. A059201, A316978, A319559, A319564, A326939, A326941, A326949.

Sequence in context: A047588 A213257 A039213 * A256450 A119605 A144146

Adjacent sequences:  A326944 A326945 A326946 * A326948 A326949 A326950

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 08 2019

STATUS

approved

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Last modified April 9 14:04 EDT 2020. Contains 333353 sequences. (Running on oeis4.)