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

0, 1, 2, 5, 6, 8, 17, 24, 34, 40, 52, 69, 70, 81, 84, 85, 88, 98, 100, 102, 104, 112, 116, 120, 128, 257, 384, 514, 640, 772, 1029, 1030, 1281, 1284, 1285, 1408, 1538, 1540, 1542, 1664, 1792, 1796, 1920, 2056, 2176, 2320, 2592, 2880, 3120, 3152, 3168, 3184

Offset

1,3

Comments

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.

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.

Links

Table of n, a(n) for n=1..52.

Example

The sequence of all set-systems that are pairwise intersecting, cointersecting, and costrict, together with their BII-numbers, begins:
    0: {}
    1: {{1}}
    2: {{2}}
    5: {{1},{1,2}}
    6: {{2},{1,2}}
    8: {{3}}
   17: {{1},{1,3}}
   24: {{3},{1,3}}
   34: {{2},{2,3}}
   40: {{3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
   69: {{1},{1,2},{1,2,3}}
   70: {{2},{1,2},{1,2,3}}
   81: {{1},{1,3},{1,2,3}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   88: {{3},{1,3},{1,2,3}}
   98: {{2},{2,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}

Mathematica

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[0, 10000], UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[bpe/@bpe[#], Intersection[#1, #2]=={}&]&&stableQ[dual[bpe/@bpe[#]], Intersection[#1, #2]=={}&]&]

Crossrefs

Equals the intersection of A326947, A326910, and A326853.

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

The non-T_0 version is A327061.

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

Sequence in context: A191204 A191140 A271430 * A097685 A136369 A305511

Adjacent sequences: A326851 A326852 A326853 * A326855 A326856 A326857

Keyword

nonn

Author

Gus Wiseman, Aug 18 2019

Status

approved