A327061 - OEIS

%I #6 Aug 19 2019 08:50:45

%S 0,1,2,4,5,6,8,16,17,24,32,34,40,52,64,65,66,68,69,70,72,80,81,84,85,

%T 88,96,98,100,102,104,112,116,120,128,256,257,384,512,514,640,772,

%U 1024,1025,1026,1028,1029,1030,1152,1280,1281,1284,1285,1408,1536,1538

%N BII-numbers of pairwise intersecting set-systems where every two covered 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 also 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 pairwise intersecting, cointersecting set-systems together with their BII-numbers begins:

%e    0: {}

%e    1: {{1}}

%e    2: {{2}}

%e    4: {{1,2}}

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

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

%e    8: {{3}}

%e   16: {{1,3}}

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

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

%e   32: {{2,3}}

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

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

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

%e   64: {{1,2,3}}

%e   65: {{1},{1,2,3}}

%e   66: {{2},{1,2,3}}

%e   68: {{1,2},{1,2,3}}

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

%e   70: {{2},{1,2},{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[0,100],stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&&stableQ[dual[bpe/@bpe[#]],Intersection[#1,#2]=={}&]&]

%Y The unlabeled multiset partition version is A319765.

%Y Equals the intersection of A326853 and A326910.

%Y The T_0 version is A326854.

%Y These set-systems are counted by A327037 (covering) and A327038 (not covering).

%Y Cf. A029931, A048793, A051185, A305843, A319774, A326031, A327039, A327040, A327041, A327057.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 18 2019