%I #6 Jul 31 2019 08:48:32
%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 80,81,82,84,85,86,87,88,89,92,93,96,97,98,100,101,102,103,104,106,
%U 108,110,112,113,114,116,117,118,119,120,121,122,124,125,126,127,128
%N BII-numbers of set-systems that are closed under union.
%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. Elements of a set-system are sometimes called edges.
%C The enumeration of these set-systems by number of covered vertices is A102896.
%e The sequence of all set-systems that are closed under union 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}}
%e 70: {{2},{1,2},{1,2,3}}
%e 71: {{1},{2},{1,2},{1,2,3}}
%e 72: {{3},{1,2,3}}
%e 76: {{1,2},{3},{1,2,3}}
%e 80: {{1,3},{1,2,3}}
%e 81: {{1},{1,3},{1,2,3}}
%e 82: {{2},{1,3},{1,2,3}}
%e 84: {{1,2},{1,3},{1,2,3}}
%e 85: {{1},{1,2},{1,3},{1,2,3}}
%e 86: {{2},{1,2},{1,3},{1,2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union@@@Tuples[bpe/@bpe[#],2]]&]
%Y Cf. A006126, A048793, A102894, A102896, A102897, A326031, A326872, A326874, A326876, A326880.
%K nonn
%O 1,3
%A _Gus Wiseman_, Jul 29 2019