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A326704 BII-numbers of antichains of nonempty sets. 46
0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 292, 304, 308, 320, 512, 513, 516, 520, 521, 524 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number 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 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, it follows that the BII-number of {{2},{1,3}} is 18.
Elements of a set-system are sometimes called edges. In an antichain of sets, no edge is a subset or superset of any other edge.
LINKS
EXAMPLE
The sequence of all antichains of nonempty sets together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
16: {{1,3}}
18: {{2},{1,3}}
20: {{1,2},{1,3}}
32: {{2,3}}
33: {{1},{2,3}}
36: {{1,2},{2,3}}
48: {{1,3},{2,3}}
52: {{1,2},{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[100], stableQ[bpe/@bpe[#], SubsetQ]&]
CROSSREFS
Antichains of sets are counted by A000372.
Antichains of nonempty sets are counted by A014466.
MM-numbers of antichains of multisets are A316476.
BII-numbers of chains of nonempty sets are A326703.
Sequence in context: A047453 A037467 A165564 * A309314 A309326 A326701
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 21 2019
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)