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 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. A minimal cover is a set-system where every edge contains at least one vertex that does not belong to any other edge.
EXAMPLE
The sequence of all minimal covers 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}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 1000], And@@Table[Union@@Delete[bpe/@bpe[#], i]!=Union@@bpe/@bpe[#], {i, Length[bpe/@bpe[#]]}]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 23 2019
STATUS
approved