cp's OEIS Frontend

A327080 BII-numbers of maximal uniform set-systems (or complete hypergraphs).

%I A327080 #5 Aug 22 2019 20:40:49
%S A327080 0,1,2,3,4,8,9,10,11,16,32,52,64,128,129,130,131,136,137,138,139,256,
%T A327080 512,772,1024,2048,2320,2592,2868,4096,8192,13376,16384,32768,32769,
%U A327080 32770,32771,32776,32777,32778,32779,32896,32897,32898,32899,32904,32905,32906
%N A327080 BII-numbers of maximal uniform set-systems (or complete hypergraphs).
%C A327080 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 A327080 A set-system is uniform if all edges have the same size.
%e A327080 The sequence of all maximal uniform set-systems together with their BII-numbers begins:
%e A327080     0: {}
%e A327080     1: {{1}}
%e A327080     2: {{2}}
%e A327080     3: {{1},{2}}
%e A327080     4: {{1,2}}
%e A327080     8: {{3}}
%e A327080     9: {{1},{3}}
%e A327080    10: {{2},{3}}
%e A327080    11: {{1},{2},{3}}
%e A327080    16: {{1,3}}
%e A327080    32: {{2,3}}
%e A327080    52: {{1,2},{1,3},{2,3}}
%e A327080    64: {{1,2,3}}
%e A327080   128: {{4}}
%e A327080   129: {{1},{4}}
%e A327080   130: {{2},{4}}
%e A327080   131: {{1},{2},{4}}
%e A327080   136: {{3},{4}}
%e A327080   137: {{1},{3},{4}}
%e A327080   138: {{2},{3},{4}}
%t A327080 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327080 Select[Range[0,100],With[{sys=bpe/@bpe[#]},#==0||SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys],Length[First[sys]]]]&]
%Y A327080 BII-numbers of uniform set-systems are A326783.
%Y A327080 The normal case (where the edges cover an initial interval) is A327081.
%Y A327080 Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.
%K A327080 nonn
%O A327080 1,3
%A A327080 _Gus Wiseman_, Aug 20 2019