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A326031
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Weight of the set-system with BII-number n.
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129
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0, 1, 1, 2, 2, 3, 3, 4, 1, 2, 2, 3, 3, 4, 4, 5, 2, 3, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5, 5, 6, 6, 7, 4, 5, 5, 6, 6, 7, 7, 8, 5, 6, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 6, 7, 4, 5, 5, 6, 6, 7, 7, 8, 5, 6, 6, 7, 7, 8, 8, 9
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OFFSET
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0,4
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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 of positive integers 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. The weight of a set-system is the sum of sizes of its elements (sometimes called its edges).
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LINKS
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FORMULA
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a(2^x + ... + 2^z) = w(x + 1) + ... + w(z + 1), where x...z are distinct nonnegative integers and w = A000120. For example, a(6) = a(2^2 + 2^1) = w(3) + w(2) = 3.
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EXAMPLE
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The sequence of set-systems together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
7: {{1},{2},{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
13: {{1},{1,2},{3}}
14: {{2},{1,2},{3}}
15: {{1},{2},{1,2},{3}}
16: {{1,3}}
17: {{1},{1,3}}
18: {{2},{1,3}}
19: {{1},{2},{1,3}}
20: {{1,2},{1,3}}
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MATHEMATICA
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Join@@bpe/@bpe[n]], {n, 0, 100}]
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CROSSREFS
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Cf. A000120, A029931, A048793, A061775, A070939, A072639, A116549, A302242, A305830, A326701, A326702, A326703, 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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