cp's OEIS Frontend

A327016 BII-numbers of finite T_0 topologies without their empty set.

%I A327016 #6 Aug 15 2019 07:30:11
%S A327016 0,1,2,5,6,7,8,17,24,25,34,40,42,69,70,71,81,85,87,88,89,93,98,102,
%T A327016 103,104,106,110,120,121,122,127,128,257,384,385,514,640,642,1029,
%U A327016 1030,1031,1281,1285,1287,1408,1409,1413,1538,1542,1543,1664,1666,1670,1920
%N A327016 BII-numbers of finite T_0 topologies without their empty set.
%C A327016 A set-system is a finite set of finite nonempty sets. The dual of a set-system 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_0 condition means that the dual is strict (no repeated edges).
%C A327016 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.
%H A327016 Gus Wiseman, <a href="/A327016/a327016.png">The first 100 finite T_0 topologies, represented as posets.</a>
%e A327016 The sequence of all finite T_0 topologies without their empty set together with their BII-numbers begins:
%e A327016    0: {}
%e A327016    1: {{1}}
%e A327016    2: {{2}}
%e A327016    5: {{1},{1,2}}
%e A327016    6: {{2},{1,2}}
%e A327016    7: {{1},{2},{1,2}}
%e A327016    8: {{3}}
%e A327016   17: {{1},{1,3}}
%e A327016   24: {{3},{1,3}}
%e A327016   25: {{1},{3},{1,3}}
%e A327016   34: {{2},{2,3}}
%e A327016   40: {{3},{2,3}}
%e A327016   42: {{2},{3},{2,3}}
%e A327016   69: {{1},{1,2},{1,2,3}}
%e A327016   70: {{2},{1,2},{1,2,3}}
%e A327016   71: {{1},{2},{1,2},{1,2,3}}
%e A327016   81: {{1},{1,3},{1,2,3}}
%e A327016   85: {{1},{1,2},{1,3},{1,2,3}}
%e A327016   87: {{1},{2},{1,2},{1,3},{1,2,3}}
%e A327016   88: {{3},{1,3},{1,2,3}}
%t A327016 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327016 Select[Range[0,1000],UnsameQ@@dual[bpe/@bpe[#]]&&SubsetQ[bpe/@bpe[#],Union[Union@@@Tuples[bpe/@bpe[#],2],DeleteCases[Intersection@@@Tuples[bpe/@bpe[#],2],{}]]]&]
%Y A327016 T_0 topologies are A001035, with unlabeled version A000112.
%Y A327016 BII-numbers of topologies without their empty set are A326876.
%Y A327016 BII-numbers of T_0 set-systems are A326947.
%Y A327016 Cf. A001930, A048793, A306445, A316978, A319564, A326031, A326872, A326875, A326939, A326941, A326959.
%K A327016 nonn
%O A327016 1,3
%A A327016 _Gus Wiseman_, Aug 14 2019