cp's OEIS Frontend

A326910 BII-numbers of pairwise intersecting set-systems.

%I A326910 #6 Aug 05 2019 07:36:49
%S A326910 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 A326910 70,72,80,81,84,85,88,96,98,100,102,104,112,116,120,128,256,257,260,
%U A326910 261,272,273,276,277,320,321,324,325,336,337,340,341,384,512,514
%N A326910 BII-numbers of pairwise intersecting set-systems.
%C A326910 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 A326910 The sequence of all pairwise intersecting set-systems together with their BII-numbers begins:
%e A326910    0: {}
%e A326910    1: {{1}}
%e A326910    2: {{2}}
%e A326910    4: {{1,2}}
%e A326910    5: {{1},{1,2}}
%e A326910    6: {{2},{1,2}}
%e A326910    8: {{3}}
%e A326910   16: {{1,3}}
%e A326910   17: {{1},{1,3}}
%e A326910   20: {{1,2},{1,3}}
%e A326910   21: {{1},{1,2},{1,3}}
%e A326910   24: {{3},{1,3}}
%e A326910   32: {{2,3}}
%e A326910   34: {{2},{2,3}}
%e A326910   36: {{1,2},{2,3}}
%e A326910   38: {{2},{1,2},{2,3}}
%e A326910   40: {{3},{2,3}}
%e A326910   48: {{1,3},{2,3}}
%e A326910   52: {{1,2},{1,3},{2,3}}
%e A326910   56: {{3},{1,3},{2,3}}
%t A326910 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326910 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326910 Select[Range[0,100],stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&]
%Y A326910 Intersecting set systems are A051185 (not-covering) or A305843 (covering).
%Y A326910 BII-numbers of set-systems with empty intersection are A326911.
%Y A326910 Cf. A006058, A048793, A326031, A326875, A326912, A326913.
%K A326910 nonn
%O A326910 1,3
%A A326910 _Gus Wiseman_, Aug 04 2019