

A326979


BIInumbers of T_1 setsystems.


17



0, 1, 2, 3, 7, 8, 9, 10, 11, 15, 25, 27, 30, 31, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 79, 91, 94, 95, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 135, 136, 137, 138, 139, 143, 153, 155, 158, 159
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OFFSET

1,3


COMMENTS

A setsystem is a finite set of finite nonempty sets. The dual of a setsystem has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_1 condition means that the dual is a (strict) antichain, meaning that none of its edges is a subset of any other.
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



EXAMPLE

The sequence of all T_1 setsystems together with their BIInumbers begins:
0: {}
1: {{1}}
2: {{2}}
3: {{1},{2}}
7: {{1},{2},{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
15: {{1},{2},{1,2},{3}}
25: {{1},{3},{1,3}}
27: {{1},{2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
31: {{1},{2},{1,2},{3},{1,3}}
42: {{2},{3},{2,3}}
43: {{1},{2},{3},{2,3}}
45: {{1},{1,2},{3},{2,3}}
47: {{1},{2},{1,2},{3},{2,3}}
51: {{1},{2},{1,3},{2,3}}
52: {{1,2},{1,3},{2,3}}


MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[0, 100], UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[dual[bpe/@bpe[#]], SubsetQ]&]


CROSSREFS

BIInumbers of T_0 setsystems are A326947.
BIInumbers of setsystems whose dual is a weak antichain are A326966.


KEYWORD

nonn


AUTHOR



STATUS

approved



