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%I #5 Sep 01 2019 08:40:41
%S 5,6,7,13,14,15,17,19,20,22,24,25,26,27,28,30,34,35,36,37,40,41,42,43,
%T 44,45,48,49,50,51,52,65,66,67,68,72,73,74,75,76,80,82,96,97,133,134,
%U 135,141,142,143,145,147,148,150,152,153,154,155,156,158,162
%N BII-numbers of set-systems with maximum degree 2.
%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 In a set-system, the degree of a vertex is the number of edges containing it.
%e The sequence of all set-systems with maximum degree 2 together with their BII-numbers begins:
%e 5: {{1},{1,2}}
%e 6: {{2},{1,2}}
%e 7: {{1},{2},{1,2}}
%e 13: {{1},{1,2},{3}}
%e 14: {{2},{1,2},{3}}
%e 15: {{1},{2},{1,2},{3}}
%e 17: {{1},{1,3}}
%e 19: {{1},{2},{1,3}}
%e 20: {{1,2},{1,3}}
%e 22: {{2},{1,2},{1,3}}
%e 24: {{3},{1,3}}
%e 25: {{1},{3},{1,3}}
%e 26: {{2},{3},{1,3}}
%e 27: {{1},{2},{3},{1,3}}
%e 28: {{1,2},{3},{1,3}}
%e 30: {{2},{1,2},{3},{1,3}}
%e 34: {{2},{2,3}}
%e 35: {{1},{2},{2,3}}
%e 36: {{1,2},{2,3}}
%e 37: {{1},{1,2},{2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[0,100],If[#==0,0,Max@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]]==2&]
%Y Positions of 2's in A327104.
%Y Graphs with maximum degree 2 are counted by A136284.
%Y Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326786, A327041, A327103.
%K nonn
%O 1,1
%A _Gus Wiseman_, Aug 26 2019
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