

A326910


BIInumbers of pairwise intersecting setsystems.


14



0, 1, 2, 4, 5, 6, 8, 16, 17, 20, 21, 24, 32, 34, 36, 38, 40, 48, 52, 56, 64, 65, 66, 68, 69, 70, 72, 80, 81, 84, 85, 88, 96, 98, 100, 102, 104, 112, 116, 120, 128, 256, 257, 260, 261, 272, 273, 276, 277, 320, 321, 324, 325, 336, 337, 340, 341, 384, 512, 514
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OFFSET

1,3


COMMENTS

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 finite set of finite nonempty sets 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. Elements of a setsystem are sometimes called edges.


LINKS

Table of n, a(n) for n=1..60.


EXAMPLE

The sequence of all pairwise intersecting 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}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
24: {{3},{1,3}}
32: {{2,3}}
34: {{2},{2,3}}
36: {{1,2},{2,3}}
38: {{2},{1,2},{2,3}}
40: {{3},{2,3}}
48: {{1,3},{2,3}}
52: {{1,2},{1,3},{2,3}}
56: {{3},{1,3},{2,3}}


MATHEMATICA

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]=={}&]&]


CROSSREFS

Intersecting set systems are A051185 (notcovering) or A305843 (covering).
BIInumbers of setsystems with empty intersection are A326911.
Cf. A006058, A048793, A326031, A326875, A326912, A326913.
Sequence in context: A325680 A176654 A185867 * A326905 A327061 A326913
Adjacent sequences: A326907 A326908 A326909 * A326911 A326912 A326913


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 04 2019


STATUS

approved



