A326911 - OEIS

%I #5 Aug 05 2019 07:36:57

%S 0,3,7,9,10,11,12,13,14,15,18,19,22,23,25,26,27,28,29,30,31,33,35,37,

%T 39,41,42,43,44,45,46,47,49,50,51,52,53,54,55,57,58,59,60,61,62,63,67,

%U 71,73,74,75,76,77,78,79,82,83,86,87,89,90,91,92,93,94,95

%N BII-numbers of set-systems with empty intersection.

%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.

%e The sequence of all set-systems with empty intersection together with their BII-numbers begins:

%e    0: {}

%e    3: {{1},{2}}

%e    7: {{1},{2},{1,2}}

%e    9: {{1},{3}}

%e   10: {{2},{3}}

%e   11: {{1},{2},{3}}

%e   12: {{1,2},{3}}

%e   13: {{1},{1,2},{3}}

%e   14: {{2},{1,2},{3}}

%e   15: {{1},{2},{1,2},{3}}

%e   18: {{2},{1,3}}

%e   19: {{1},{2},{1,3}}

%e   22: {{2},{1,2},{1,3}}

%e   23: {{1},{2},{1,2},{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   29: {{1},{1,2},{3},{1,3}}

%e   30: {{2},{1,2},{3},{1,3}}

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

%t Select[Range[0,100],#==0||Intersection@@bpe/@bpe[#]=={}&]

%Y Cf. A048793, A051185, A305843, A317752, A317755, A317757, A319077, A326031, A326910, A326912.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 04 2019