%I #5 Aug 22 2019 20:40:57

%S 1,3,4,11,52,64,139,2868,13376,16384,32907

%N BII-numbers of maximal uniform set-systems covering an initial interval of positive integers.

%C 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.

%C A set-system is uniform if all edges have the same size.

%e The sequence of all maximal uniform set-systems covering an initial interval together with their BII-numbers begins:

%e 0: {}

%e 1: {{1}}

%e 3: {{1},{2}}

%e 4: {{1,2}}

%e 11: {{1},{2},{3}}

%e 52: {{1,2},{1,3},{2,3}}

%e 64: {{1,2,3}}

%e 139: {{1},{2},{3},{4}}

%e 2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

%e 13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

%e 16384: {{1,2,3,4}}

%e 32907: {{1},{2},{3},{4},{5}}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];

%t Select[Range[1000],With[{sys=bpe/@bpe[#]},#==0||normQ[Union@@sys]&&SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys],Length[First[sys]]]]&]

%Y BII-numbers of uniform set-systems are A326783.

%Y BII-numbers of maximal uniform set-systems are A327080.

%Y Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.

%K nonn,more

%O 1,2

%A _Gus Wiseman_, Aug 20 2019