A326854 - OEIS

%I #4 Aug 18 2019 11:27:55

%S 0,1,2,5,6,8,17,24,34,40,52,69,70,81,84,85,88,98,100,102,104,112,116,

%T 120,128,257,384,514,640,772,1029,1030,1281,1284,1285,1408,1538,1540,

%U 1542,1664,1792,1796,1920,2056,2176,2320,2592,2880,3120,3152,3168,3184

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

%C 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 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 The sequence of all set-systems that are pairwise intersecting, cointersecting, and costrict, together with their BII-numbers, begins:

%e     0: {}

%e     1: {{1}}

%e     2: {{2}}

%e     5: {{1},{1,2}}

%e     6: {{2},{1,2}}

%e     8: {{3}}

%e    17: {{1},{1,3}}

%e    24: {{3},{1,3}}

%e    34: {{2},{2,3}}

%e    40: {{3},{2,3}}

%e    52: {{1,2},{1,3},{2,3}}

%e    69: {{1},{1,2},{1,2,3}}

%e    70: {{2},{1,2},{1,2,3}}

%e    81: {{1},{1,3},{1,2,3}}

%e    84: {{1,2},{1,3},{1,2,3}}

%e    85: {{1},{1,2},{1,3},{1,2,3}}

%e    88: {{3},{1,3},{1,2,3}}

%e    98: {{2},{2,3},{1,2,3}}

%e   100: {{1,2},{2,3},{1,2,3}}

%e   102: {{2},{1,2},{2,3},{1,2,3}}

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%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[0,10000],UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&&stableQ[dual[bpe/@bpe[#]],Intersection[#1,#2]=={}&]&]

%Y Equals the intersection of A326947, A326910, and A326853.

%Y These set-systems are counted by A319774 (covering).

%Y The non-T_0 version is A327061.

%Y Cf. A029931, A048793, A051185, A305843, A319765, A326031, A327037, A327038, A327041, A327052, A327053.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 18 2019