A326913 - OEIS

%I #6 Aug 05 2019 07:37:12

%S 0,1,2,4,5,6,8,16,17,24,32,34,40,64,65,66,68,69,70,72,80,81,85,88,96,

%T 98,102,104,120,128,256,257,384,512,514,640,1024,1025,1026,1028,1029,

%U 1030,1152,1280,1281,1285,1408,1536,1538,1542,1664,1920,2048,2056,2176

%N BII-numbers of set-systems (without {}) closed under union and intersection.

%C A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

%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 closed under union and intersection together with their BII-numbers begins:

%e    0: {}

%e    1: {{1}}

%e    2: {{2}}

%e    4: {{1,2}}

%e    5: {{1},{1,2}}

%e    6: {{2},{1,2}}

%e    8: {{3}}

%e   16: {{1,3}}

%e   17: {{1},{1,3}}

%e   24: {{3},{1,3}}

%e   32: {{2,3}}

%e   34: {{2},{2,3}}

%e   40: {{3},{2,3}}

%e   64: {{1,2,3}}

%e   65: {{1},{1,2,3}}

%e   66: {{2},{1,2,3}}

%e   68: {{1,2},{1,2,3}}

%e   69: {{1},{1,2},{1,2,3}}

%e   70: {{2},{1,2},{1,2,3}}

%e   72: {{3},{1,2,3}}

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

%t Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union@@@Tuples[bpe/@bpe[#],2]]&&SubsetQ[bpe/@bpe[#],Intersection@@@Tuples[bpe/@bpe[#],2]]&]

%Y Cf. A048793, A102894, A102895, A102896, A102897, A326031, A326875, A326876, A326878, A326880, A326901.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 04 2019