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A327081 BII-numbers of maximal uniform set-systems covering an initial interval of positive integers.

%I A327081 #5 Aug 22 2019 20:40:57
%S A327081 1,3,4,11,52,64,139,2868,13376,16384,32907
%N A327081 BII-numbers of maximal uniform set-systems covering an initial interval of positive integers.
%C A327081 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 A327081 A set-system is uniform if all edges have the same size.
%e A327081 The sequence of all maximal uniform set-systems covering an initial interval together with their BII-numbers begins:
%e A327081       0: {}
%e A327081       1: {{1}}
%e A327081       3: {{1},{2}}
%e A327081       4: {{1,2}}
%e A327081      11: {{1},{2},{3}}
%e A327081      52: {{1,2},{1,3},{2,3}}
%e A327081      64: {{1,2,3}}
%e A327081     139: {{1},{2},{3},{4}}
%e A327081    2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
%e A327081   13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
%e A327081   16384: {{1,2,3,4}}
%e A327081   32907: {{1},{2},{3},{4},{5}}
%t A327081 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327081 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A327081 Select[Range[1000],With[{sys=bpe/@bpe[#]},#==0||normQ[Union@@sys]&&SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys],Length[First[sys]]]]&]
%Y A327081 BII-numbers of uniform set-systems are A326783.
%Y A327081 BII-numbers of maximal uniform set-systems are A327080.
%Y A327081 Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.
%K A327081 nonn,more
%O A327081 1,2
%A A327081 _Gus Wiseman_, Aug 20 2019