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%I #5 Aug 05 2019 07:37:04
%S 0,52,116,772,832,836,1072,1076,1136,1140,1796,1856,1860,2320,2368,
%T 2384,2592,2624,2656,2880,3088,3104,3120,3136,3152,3168,3184,3344,
%U 3392,3408,3616,3648,3680,3904,4132,4148,4196,4212,4612,4640,4644,4672,4676,4704,4708
%N BII-numbers of pairwise intersecting set-systems with empty intersection.
%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 with empty intersection, together with their BII-numbers, begins:
%e 0: {}
%e 52: {{1,2},{1,3},{2,3}}
%e 116: {{1,2},{1,3},{2,3},{1,2,3}}
%e 772: {{1,2},{1,4},{2,4}}
%e 832: {{1,2,3},{1,4},{2,4}}
%e 836: {{1,2},{1,2,3},{1,4},{2,4}}
%e 1072: {{1,3},{2,3},{1,2,4}}
%e 1076: {{1,2},{1,3},{2,3},{1,2,4}}
%e 1136: {{1,3},{2,3},{1,2,3},{1,2,4}}
%e 1140: {{1,2},{1,3},{2,3},{1,2,3},{1,2,4}}
%e 1796: {{1,2},{1,4},{2,4},{1,2,4}}
%e 1856: {{1,2,3},{1,4},{2,4},{1,2,4}}
%e 1860: {{1,2},{1,2,3},{1,4},{2,4},{1,2,4}}
%e 2320: {{1,3},{1,4},{3,4}}
%e 2368: {{1,2,3},{1,4},{3,4}}
%e 2384: {{1,3},{1,2,3},{1,4},{3,4}}
%e 2592: {{2,3},{2,4},{3,4}}
%e 2624: {{1,2,3},{2,4},{3,4}}
%e 2656: {{2,3},{1,2,3},{2,4},{3,4}}
%e 2880: {{1,2,3},{1,4},{2,4},{3,4}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[0,1000],(#==0||Intersection@@bpe/@bpe[#]=={})&&stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&]
%Y Cf. A048793, A051185, A305843, A317752, A317755, A317757, A319077, A326031, A326910, A326911.
%Y Cf. A318715, A319759, A319762, A319763, A319764.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 04 2019
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