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A326854 BII-numbers of T_0 (costrict), pairwise intersecting set-systems where every two vertices appear together in some edge (cointersecting).

%I A326854 #4 Aug 18 2019 11:27:55
%S A326854 0,1,2,5,6,8,17,24,34,40,52,69,70,81,84,85,88,98,100,102,104,112,116,
%T A326854 120,128,257,384,514,640,772,1029,1030,1281,1284,1285,1408,1538,1540,
%U A326854 1542,1664,1792,1796,1920,2056,2176,2320,2592,2880,3120,3152,3168,3184
%N A326854 BII-numbers of T_0 (costrict), pairwise intersecting set-systems where every two vertices appear together in some edge (cointersecting).
%C A326854 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 pairwise intersecting set-systems whose dual is strict and pairwise intersecting.
%C A326854 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 A326854 The sequence of all set-systems that are pairwise intersecting, cointersecting, and costrict, together with their BII-numbers, begins:
%e A326854     0: {}
%e A326854     1: {{1}}
%e A326854     2: {{2}}
%e A326854     5: {{1},{1,2}}
%e A326854     6: {{2},{1,2}}
%e A326854     8: {{3}}
%e A326854    17: {{1},{1,3}}
%e A326854    24: {{3},{1,3}}
%e A326854    34: {{2},{2,3}}
%e A326854    40: {{3},{2,3}}
%e A326854    52: {{1,2},{1,3},{2,3}}
%e A326854    69: {{1},{1,2},{1,2,3}}
%e A326854    70: {{2},{1,2},{1,2,3}}
%e A326854    81: {{1},{1,3},{1,2,3}}
%e A326854    84: {{1,2},{1,3},{1,2,3}}
%e A326854    85: {{1},{1,2},{1,3},{1,2,3}}
%e A326854    88: {{3},{1,3},{1,2,3}}
%e A326854    98: {{2},{2,3},{1,2,3}}
%e A326854   100: {{1,2},{2,3},{1,2,3}}
%e A326854   102: {{2},{1,2},{2,3},{1,2,3}}
%t A326854 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326854 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326854 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326854 Select[Range[0,10000],UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&&stableQ[dual[bpe/@bpe[#]],Intersection[#1,#2]=={}&]&]
%Y A326854 Equals the intersection of A326947, A326910, and A326853.
%Y A326854 These set-systems are counted by A319774 (covering).
%Y A326854 The non-T_0 version is A327061.
%Y A326854 Cf. A029931, A048793, A051185, A305843, A319765, A326031, A327037, A327038, A327041, A327052, A327053.
%K A326854 nonn
%O A326854 1,3
%A A326854 _Gus Wiseman_, Aug 18 2019