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BII-numbers of set-systems that are closed under nonempty intersection.
12

%I #6 Jul 31 2019 10:00:42

%S 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,21,23,24,25,26,27,

%T 29,31,32,33,34,35,38,39,40,41,42,43,46,47,56,57,58,59,63,64,65,66,67,

%U 68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,85,87,88

%N BII-numbers of set-systems that are closed under nonempty 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.

%C The enumeration of these set-systems by number of covered vertices is A326881.

%e Most small numbers are in the sequence, but the sequence of non-terms together with the set-systems with those BII-numbers begins:

%e 20: {{1,2},{1,3}}

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

%e 28: {{1,2},{3},{1,3}}

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

%e 36: {{1,2},{2,3}}

%e 37: {{1},{1,2},{2,3}}

%e 44: {{1,2},{3},{2,3}}

%e 45: {{1},{1,2},{3},{2,3}}

%e 48: {{1,3},{2,3}}

%e 49: {{1},{1,3},{2,3}}

%e 50: {{2},{1,3},{2,3}}

%e 51: {{1},{2},{1,3},{2,3}}

%e 52: {{1,2},{1,3},{2,3}}

%e 53: {{1},{1,2},{1,3},{2,3}}

%e 54: {{2},{1,2},{1,3},{2,3}}

%e 55: {{1},{2},{1,2},{1,3},{2,3}}

%e 60: {{1,2},{3},{1,3},{2,3}}

%e 61: {{1},{1,2},{3},{1,3},{2,3}}

%e 62: {{2},{1,2},{3},{1,3},{2,3}}

%e 84: {{1,2},{1,3},{1,2,3}}

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

%t Select[Range[0,100],SubsetQ[bpe/@bpe[#],Intersection@@@Select[Tuples[bpe/@bpe[#],2],Intersection@@#!={}&]]&]

%Y Cf. A006126, A048793, A102894, A102895, A102896, A102897, A306445, A326031, A326872, A326874, A326875, A326876, A326881.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 29 2019