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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). 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 30 11:55 EDT 2020. Contains 337439 sequences. (Running on oeis4.)