
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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem (finite set of finite nonempty sets of positive integers) has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.
A setsystem is an antichain if no edge is a proper subset of any other.


EXAMPLE

The sequence of terms together with their corresponding setsystems begins:
0: {}
1: {{1}}
3: {{1},{2}}
11: {{1},{2},{3}}
139: {{1},{2},{3},{4}}
820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}


MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_]:=!Apply[Or, Outer[#1=!=#2&&SubsetQ[#1, #2]&, u, u, 1], {0, 1}];
First/@GatherBy[Select[Range[0, 10000], stableQ[bpe/@bpe[#]]&], Length[bpe[#]]&]
