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 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, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
A set-system is an antichain if no edge is a proper subset of any other.
Empty intersection means there is no vertex in common to all the edges
EXAMPLE
The sequence of terms together with their binary expansions and corresponding set-systems begins:
0: 0 ~ {}
3: 11 ~ {{1},{2}}
9: 1001 ~ {{1},{3}}
10: 1010 ~ {{2},{3}}
11: 1011 ~ {{1},{2},{3}}
12: 1100 ~ {{1,2},{3}}
18: 10010 ~ {{2},{1,3}}
33: 100001 ~ {{1},{2,3}}
52: 110100 ~ {{1,2},{1,3},{2,3}}
129: 10000001 ~ {{1},{4}}
130: 10000010 ~ {{2},{4}}
131: 10000011 ~ {{1},{2},{4}}
132: 10000100 ~ {{1,2},{4}}
136: 10001000 ~ {{3},{4}}
137: 10001001 ~ {{1},{3},{4}}
138: 10001010 ~ {{2},{3},{4}}
139: 10001011 ~ {{2},{3},{4}}
140: 10001100 ~ {{1,2},{3},{4}}
144: 10010000 ~ {{1,3},{4}}
146: 10010010 ~ {{2},{1,3},{4}}
148: 10010100 ~ {{1,2},{1,3},{4}}
MATHEMATICA
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], #==0||Intersection@@bpe/@bpe[#]=={}&&stableQ[bpe/@bpe[#], SubsetQ]&]
CROSSREFS
BII-numbers of intersecting set-systems with empty intersecting are A326912.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 28 2019
STATUS
approved