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%I #5 Jul 27 2019 14:57:51
%S 0,1,2,3,4,8,9,10,11,16,32,52,64,128,129,130,131,136,137,138,139,256,
%T 288,512,528,772,816,1024,2048,2052,2320,2340,2580,2592,2868,4096,
%U 8192,13376,16384,32768,32769,32770,32771,32776,32777,32778,32779,32896,32897
%N BII-numbers of uniform 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. Every 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. A set-system is uniform if all edges have the same size, and regular if all vertices appear the same number of times.
%F Intersection of A326783 and A326784.
%e The sequence of all uniform 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 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[0,1000],SameQ@@Length/@bpe/@bpe[#]&&SameQ@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]&]
%Y Cf. A000120, A029931, A048793, A070939, A319056, A319189, A321698, A326031, A326701, A326783 (uniform), A326784 (regular), A326788.
%K nonn
%O 1,3
%A _Gus Wiseman_, Jul 25 2019
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