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%I #6 Aug 05 2019 07:36:49
%S 0,1,2,4,5,6,8,16,17,20,21,24,32,34,36,38,40,48,52,56,64,65,66,68,69,
%T 70,72,80,81,84,85,88,96,98,100,102,104,112,116,120,128,256,257,260,
%U 261,272,273,276,277,320,321,324,325,336,337,340,341,384,512,514
%N BII-numbers of pairwise intersecting set-systems.
%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.
%e The sequence of all pairwise intersecting set-systems 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 8: {{3}}
%e 16: {{1,3}}
%e 17: {{1},{1,3}}
%e 20: {{1,2},{1,3}}
%e 21: {{1},{1,2},{1,3}}
%e 24: {{3},{1,3}}
%e 32: {{2,3}}
%e 34: {{2},{2,3}}
%e 36: {{1,2},{2,3}}
%e 38: {{2},{1,2},{2,3}}
%e 40: {{3},{2,3}}
%e 48: {{1,3},{2,3}}
%e 52: {{1,2},{1,3},{2,3}}
%e 56: {{3},{1,3},{2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t Select[Range[0,100],stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&]
%Y Intersecting set systems are A051185 (not-covering) or A305843 (covering).
%Y BII-numbers of set-systems with empty intersection are A326911.
%Y Cf. A006058, A048793, A326031, A326875, A326912, A326913.
%K nonn
%O 1,3
%A _Gus Wiseman_, Aug 04 2019
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