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A326703
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BII-numbers of chains of nonempty sets.
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19
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0, 1, 2, 4, 5, 6, 8, 16, 17, 24, 32, 34, 40, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 256, 257, 384, 512, 514, 640, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1408, 1536, 1538, 1664, 2048, 2056, 2176, 4096, 4097, 4104, 4112, 4113, 4120
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OFFSET
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1,3
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COMMENTS
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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 a chain of sets, every edge is a subset or superset of every other edge.
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LINKS
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EXAMPLE
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The sequence of all chains of nonempty sets together with their BII-numbers 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}}
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}}
72: {{3},{1,2,3}}
80: {{1,3},{1,2,3}}
81: {{1},{1,3},{1,2,3}}
88: {{3},{1,3},{1,2,3}}
96: {{2,3},{1,2,3}}
98: {{2},{2,3},{1,2,3}}
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MATHEMATICA
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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[#], !SubsetQ[#1, #2]&&!SubsetQ[#2, #1]&]&]
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CROSSREFS
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Chains of nonempty sets are counted by A000629.
MM-numbers of chains of multisets are A318991.
BII-numbers of antichains of nonempty sets are A326704.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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