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Number of distinct vertices in the set-system with BII-number n.
45

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

%S 0,1,1,2,2,2,2,2,1,2,2,3,3,3,3,3,2,2,3,3,3,3,3,3,2,2,3,3,3,3,3,3,2,3,

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

%U 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3

%N Number of distinct vertices 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. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. 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.

%e The BII-number of {{1,2},{1,4}} is 260, with distinct vertices {1,2,4}, so a(260) = 3.

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

%t Table[Length[Union@@bpe/@bpe[n]],{n,0,100}]

%Y Positions of first appearances are A072639.

%Y Cf. A000120, A029931, A048793, A070939, A326031, A326701, A326703, A326704.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jul 22 2019