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A326853 BII-numbers of set-systems where every two covered vertices appear together in some edge (cointersecting).

%I A326853 #4 Aug 18 2019 11:26:56
%S A326853 0,1,2,4,5,6,7,8,16,17,24,25,32,34,40,42,52,53,54,55,60,61,62,63,64,
%T A326853 65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,
%U A326853 88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105
%N A326853 BII-numbers of set-systems where every two covered vertices appear together in some edge (cointersecting).
%C A326853 A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence gives all BII-numbers (defined below) of set-systems that are cointersecting, meaning their dual is pairwise intersecting.
%C A326853 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 set-system 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.
%e A326853 The sequence of all cointersecting set-systems together with their BII-numbers begins:
%e A326853    0: {}
%e A326853    1: {{1}}
%e A326853    2: {{2}}
%e A326853    4: {{1,2}}
%e A326853    5: {{1},{1,2}}
%e A326853    6: {{2},{1,2}}
%e A326853    7: {{1},{2},{1,2}}
%e A326853    8: {{3}}
%e A326853   16: {{1,3}}
%e A326853   17: {{1},{1,3}}
%e A326853   24: {{3},{1,3}}
%e A326853   25: {{1},{3},{1,3}}
%e A326853   32: {{2,3}}
%e A326853   34: {{2},{2,3}}
%e A326853   40: {{3},{2,3}}
%e A326853   42: {{2},{3},{2,3}}
%e A326853   52: {{1,2},{1,3},{2,3}}
%e A326853   53: {{1},{1,2},{1,3},{2,3}}
%e A326853   54: {{2},{1,2},{1,3},{2,3}}
%e A326853   55: {{1},{2},{1,2},{1,3},{2,3}}
%t A326853 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326853 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326853 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326853 Select[Range[0,100],stableQ[dual[bpe/@bpe[#]],Intersection[#1,#2]=={}&]&]
%Y A326853 BII-numbers of pairwise intersecting set-systems are A326910.
%Y A326853 Cointersecting set-systems are A327039, with covering version A327040.
%Y A326853 The T_0 or costrict case is A327052.
%Y A326853 Cf. A029931, A048793, A326031, A327020, A327037, A327041, A327053, A327057.
%K A326853 nonn
%O A326853 1,3
%A A326853 _Gus Wiseman_, Aug 18 2019