

A326947


BIInumbers of T_0 setsystems.


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
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OFFSET

1,3


COMMENTS

The dual of a setsystem 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 setsystem with BIInumber 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 BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.


LINKS

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


EXAMPLE

The sequence of all T_0 setsystems 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 setsystems 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



