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A309314
BII-numbers of hyperforests.
13
0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 292, 304, 320, 512, 513, 516, 520, 521, 524, 528, 532
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, the BII-number of {{2},{1,3}} is 18.
Elements of a set-system are sometimes called edges. In an antichain, no edge is a subset or superset of any other edge. A hyperforest is an antichain of nonempty sets whose connected components are hypertrees, meaning they have density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices.
EXAMPLE
The sequence of all hyperforests 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}}
64: {{1,2,3}}
128: {{4}}
129: {{1},{4}}
130: {{2},{4}}
131: {{1},{2},{4}}
132: {{1,2},{4}}
136: {{3},{4}}
137: {{1},{3},{4}}
CROSSREFS
Other BII-numbers: A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).
Sequence in context: A037467 A165564 A326704 * A309326 A326701 A061887
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 23 2019
STATUS
approved