A327103 - OEIS

%I #9 Sep 01 2019 08:41:06

%S 0,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,

%T 1,1,1,1,1,1,1,1,2,1,1,2,1,2,1,1,1,2,2,2,2,2,1,1,1,2,2,2,2,3,1,1,1,1,

%U 1,1,1,1,1,1,1,2,2,2,2,2,1,1,2,2,2,2,2,2

%N Minimum vertex-degree in 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) = 2.

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

%t Table[If[n==0,0,Min@@Length/@Split[Sort[Join@@bpe/@bpe[n]]]],{n,0,100}]

%Y The maximum vertex-degree is A327104.

%Y Positions of 1's are A327105.

%Y Positions of terms > 1 are A327107.

%Y Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326783, A326786, A327041, A327228, A327229.

%K nonn

%O 0,8

%A _Gus Wiseman_, Aug 26 2019