%I #5 Jul 27 2019 14:57:51

%S 0,1,2,3,4,7,8,9,10,11,12,16,18,25,30,32,33,42,45,51,52,63,64,75,76,

%T 82,94,97,109,115,116,127,128,129,130,131,132,136,137,138,139,140,144,

%U 146,160,161,192,256,258,264,266,288,385,390,408,427,428,434,458

%N BII-numbers of regular set-systems.

%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. A set-system is regular if all vertices appear the same number of times.

%e The sequence of all regular 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 7: {{1},{2},{1,2}}

%e 8: {{3}}

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

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

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

%e 12: {{1,2},{3}}

%e 16: {{1,3}}

%e 18: {{2},{1,3}}

%e 25: {{1},{3},{1,3}}

%e 30: {{2},{1,2},{3},{1,3}}

%e 32: {{2,3}}

%e 33: {{1},{2,3}}

%e 42: {{2},{3},{2,3}}

%e 45: {{1},{1,2},{3},{2,3}}

%e 51: {{1},{2},{1,3},{2,3}}

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

%t Select[Range[0,100],SameQ@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]&]

%Y Cf. A000120, A001511, A005176, A029931, A048793, A070939, A295193, A322554, A326031, A326701, A326783 (uniform), A326785 (uniform regular).

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 25 2019