

A327061


BIInumbers of pairwise intersecting setsystems where every two covered vertices appear together in some edge (cointersecting).


2



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, 88, 96, 98, 100, 102, 104, 112, 116, 120, 128, 256, 257, 384, 512, 514, 640, 772, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1284, 1285, 1408, 1536, 1538
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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 also 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..56.


EXAMPLE

The sequence of all pairwise intersecting, cointersecting setsystems together with their BIInumbers begins:
0: {}
1: {{1}}
2: {{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
8: {{3}}
16: {{1,3}}
17: {{1},{1,3}}
24: {{3},{1,3}}
32: {{2,3}}
34: {{2},{2,3}}
40: {{3},{2,3}}
52: {{1,2},{1,3},{2,3}}
64: {{1,2,3}}
65: {{1},{1,2,3}}
66: {{2},{1,2,3}}
68: {{1,2},{1,2,3}}
69: {{1},{1,2},{1,2,3}}
70: {{2},{1,2},{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, 100], stableQ[bpe/@bpe[#], Intersection[#1, #2]=={}&]&&stableQ[dual[bpe/@bpe[#]], Intersection[#1, #2]=={}&]&]


CROSSREFS

The unlabeled multiset partition version is A319765.
Equals the intersection of A326853 and A326910.
The T_0 version is A326854.
These setsystems are counted by A327037 (covering) and A327038 (not covering).
Cf. A029931, A048793, A051185, A305843, A319774, A326031, A327039, A327040, A327041, A327057.
Sequence in context: A185867 A326910 A326905 * A326913 A326703 A244008
Adjacent sequences: A327058 A327059 A327060 * A327062 A327063 A327064


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 18 2019


STATUS

approved



