

A326880


BIInumbers of setsystems that are closed under nonempty intersection.


12



0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 46, 47, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88
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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.
The enumeration of these setsystems by number of covered vertices is A326881.


LINKS

Table of n, a(n) for n=1..68.


EXAMPLE

Most small numbers are in the sequence, but the sequence of nonterms together with the setsystems with those BIInumbers begins:
20: {{1,2},{1,3}}
22: {{2},{1,2},{1,3}}
28: {{1,2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
36: {{1,2},{2,3}}
37: {{1},{1,2},{2,3}}
44: {{1,2},{3},{2,3}}
45: {{1},{1,2},{3},{2,3}}
48: {{1,3},{2,3}}
49: {{1},{1,3},{2,3}}
50: {{2},{1,3},{2,3}}
51: {{1},{2},{1,3},{2,3}}
52: {{1,2},{1,3},{2,3}}
53: {{1},{1,2},{1,3},{2,3}}
54: {{2},{1,2},{1,3},{2,3}}
55: {{1},{2},{1,2},{1,3},{2,3}}
60: {{1,2},{3},{1,3},{2,3}}
61: {{1},{1,2},{3},{1,3},{2,3}}
62: {{2},{1,2},{3},{1,3},{2,3}}
84: {{1,2},{1,3},{1,2,3}}


MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], SubsetQ[bpe/@bpe[#], Intersection@@@Select[Tuples[bpe/@bpe[#], 2], Intersection@@#!={}&]]&]


CROSSREFS

Cf. A006126, A048793, A102894, A102895, A102896, A102897, A306445, A326031, A326872, A326874, A326875, A326876, A326881.
Sequence in context: A320249 A309066 A084981 * A078453 A052425 A272074
Adjacent sequences: A326877 A326878 A326879 * A326881 A326882 A326883


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jul 29 2019


STATUS

approved



