

A326854


BIInumbers of T_0 (costrict), pairwise intersecting setsystems 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
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OFFSET

1,3


COMMENTS

A setsystem is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a setsystem 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 BIInumbers (defined below) of pairwise intersecting setsystems 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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18.


LINKS

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


EXAMPLE

The sequence of all setsystems that are pairwise intersecting, cointersecting, and costrict, together with their BIInumbers, 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 setsystems are counted by A319774 (covering).
The nonT_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



