OFFSET
1,2
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 set-system (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 set-system is uniform if all edges have the same size.
EXAMPLE
The sequence of all maximal uniform set-systems covering an initial interval together with their BII-numbers begins:
0: {}
1: {{1}}
3: {{1},{2}}
4: {{1,2}}
11: {{1},{2},{3}}
52: {{1,2},{1,3},{2,3}}
64: {{1,2,3}}
139: {{1},{2},{3},{4}}
2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
16384: {{1,2,3,4}}
32907: {{1},{2},{3},{4},{5}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Select[Range[1000], With[{sys=bpe/@bpe[#]}, #==0||normQ[Union@@sys]&&SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys], Length[First[sys]]]]&]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 20 2019
STATUS
approved