|
%I #8 Jul 31 2019 08:48:43
%S 0,1,2,4,5,6,7,8,16,17,24,25,32,34,40,42,64,65,66,68,69,70,71,72,76,
%T 80,81,82,85,87,88,89,93,96,97,98,102,103,104,106,110,120,121,122,127,
%U 128,256,257,384,385,512,514,640,642,1024,1025,1026,1028,1029,1030
%N BII-numbers of finite topologies without their empty set.
%C A finite topology is a finite set of finite sets closed under union and intersection and containing {} and the vertex set.
%C 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.
%C The enumeration of finite topologies by number of points is given by A000798.
%H Wikipedia <a href="https://en.wikipedia.org/wiki/Topological_space">Topological space</a>
%e The sequence of all finite topologies without their empty set together with their BII-numbers begins:
%e 0: {}
%e 1: {{1}}
%e 2: {{2}}
%e 4: {{1,2}}
%e 5: {{1},{1,2}}
%e 6: {{2},{1,2}}
%e 7: {{1},{2},{1,2}}
%e 8: {{3}}
%e 16: {{1,3}}
%e 17: {{1},{1,3}}
%e 24: {{3},{1,3}}
%e 25: {{1},{3},{1,3}}
%e 32: {{2,3}}
%e 34: {{2},{2,3}}
%e 40: {{3},{2,3}}
%e 42: {{2},{3},{2,3}}
%e 64: {{1,2,3}}
%e 65: {{1},{1,2,3}}
%e 66: {{2},{1,2,3}}
%e 68: {{1,2},{1,2,3}}
%e 69: {{1},{1,2},{1,2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union[Union@@@Tuples[bpe/@bpe[#],2],DeleteCases[Intersection@@@Tuples[bpe/@bpe[#],2],{}]]]&]
%Y Cf. A000798, A001930, A003465, A048793, A102894, A102896, A326031, A326872, A326875, A326878.
%K nonn
%O 1,3
%A _Gus Wiseman_, Jul 29 2019
|
