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 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.
EXAMPLE
The sequence of all pairwise intersecting set-systems with empty intersection, together with their BII-numbers, begins:
0: {}
52: {{1,2},{1,3},{2,3}}
116: {{1,2},{1,3},{2,3},{1,2,3}}
772: {{1,2},{1,4},{2,4}}
832: {{1,2,3},{1,4},{2,4}}
836: {{1,2},{1,2,3},{1,4},{2,4}}
1072: {{1,3},{2,3},{1,2,4}}
1076: {{1,2},{1,3},{2,3},{1,2,4}}
1136: {{1,3},{2,3},{1,2,3},{1,2,4}}
1140: {{1,2},{1,3},{2,3},{1,2,3},{1,2,4}}
1796: {{1,2},{1,4},{2,4},{1,2,4}}
1856: {{1,2,3},{1,4},{2,4},{1,2,4}}
1860: {{1,2},{1,2,3},{1,4},{2,4},{1,2,4}}
2320: {{1,3},{1,4},{3,4}}
2368: {{1,2,3},{1,4},{3,4}}
2384: {{1,3},{1,2,3},{1,4},{3,4}}
2592: {{2,3},{2,4},{3,4}}
2624: {{1,2,3},{2,4},{3,4}}
2656: {{2,3},{1,2,3},{2,4},{3,4}}
2880: {{1,2,3},{1,4},{2,4},{3,4}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 1000], (#==0||Intersection@@bpe/@bpe[#]=={})&&stableQ[bpe/@bpe[#], Intersection[#1, #2]=={}&]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 04 2019
STATUS
approved