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%I #6 Sep 01 2019 08:40:57
%S 0,1,1,1,1,2,2,2,1,1,1,1,1,2,2,2,1,2,1,2,2,3,2,3,2,2,2,2,2,3,2,3,1,1,
%T 2,2,2,2,3,3,2,2,2,2,2,2,3,3,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,1,2,2,2,
%U 2,3,3,3,2,2,2,2,2,3,3,3,2,3,2,3,3,4,3
%N Maximum vertex-degree of the set-system with BII-number n.
%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 BII-number of {{2},{3},{1,2},{1,3},{2,3}} is 62, and its degrees are (2,3,3), so a(62) = 3.
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Table[If[n==0,0,Max@@Length/@Split[Sort[Join@@bpe/@bpe[n]]]],{n,0,100}]
%Y Positions of 1's are A326701 (BII-numbers of set-partitions).
%Y The minimum vertex-degree is A327103.
%Y Positions of 2's are A327106.
%Y Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326786, A327041.
%K nonn
%O 0,6
%A _Gus Wiseman_, Aug 26 2019
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