A327080 - OEIS

%I #5 Aug 22 2019 20:40:49

%S 0,1,2,3,4,8,9,10,11,16,32,52,64,128,129,130,131,136,137,138,139,256,

%T 512,772,1024,2048,2320,2592,2868,4096,8192,13376,16384,32768,32769,

%U 32770,32771,32776,32777,32778,32779,32896,32897,32898,32899,32904,32905,32906

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

%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 together with their BII-numbers begins:

%e     0: {}

%e     1: {{1}}

%e     2: {{2}}

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

%e     4: {{1,2}}

%e     8: {{3}}

%e     9: {{1},{3}}

%e    10: {{2},{3}}

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

%e    16: {{1,3}}

%e    32: {{2,3}}

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

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

%e   128: {{4}}

%e   129: {{1},{4}}

%e   130: {{2},{4}}

%e   131: {{1},{2},{4}}

%e   136: {{3},{4}}

%e   137: {{1},{3},{4}}

%e   138: {{2},{3},{4}}

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

%t Select[Range[0,100],With[{sys=bpe/@bpe[#]},#==0||SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys],Length[First[sys]]]]&]

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

%Y The normal case (where the edges cover an initial interval) is A327081.

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 20 2019