cp's OEIS Frontend

A326905 BII-numbers of set-systems (without {}) closed under intersection.

%I A326905 #5 Aug 05 2019 07:36:41
%S A326905 0,1,2,4,5,6,8,16,17,21,24,32,34,38,40,56,64,65,66,68,69,70,72,80,81,
%T A326905 85,88,96,98,102,104,120,128,256,257,261,273,277,321,325,337,341,384,
%U A326905 512,514,518,546,550,578,582,610,614,640,896,1024,1025,1026,1028
%N A326905 BII-numbers of set-systems (without {}) closed under intersection.
%C A326905 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 A326905 The sequence of all set-systems closed under intersection together with their BII-numbers begins:
%e A326905    0: {}
%e A326905    1: {{1}}
%e A326905    2: {{2}}
%e A326905    4: {{1,2}}
%e A326905    5: {{1},{1,2}}
%e A326905    6: {{2},{1,2}}
%e A326905    8: {{3}}
%e A326905   16: {{1,3}}
%e A326905   17: {{1},{1,3}}
%e A326905   21: {{1},{1,2},{1,3}}
%e A326905   24: {{3},{1,3}}
%e A326905   32: {{2,3}}
%e A326905   34: {{2},{2,3}}
%e A326905   38: {{2},{1,2},{2,3}}
%e A326905   40: {{3},{2,3}}
%e A326905   56: {{3},{1,3},{2,3}}
%e A326905   64: {{1,2,3}}
%e A326905   65: {{1},{1,2,3}}
%e A326905   66: {{2},{1,2,3}}
%e A326905   68: {{1,2},{1,2,3}}
%t A326905 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326905 Select[Range[0,100],SubsetQ[bpe/@bpe[#],Intersection@@@Tuples[bpe/@bpe[#],2]]&]
%Y A326905 The case with union instead of intersection is A326875.
%Y A326905 The case closed under union and intersection is A326913.
%Y A326905 Set-systems closed under intersection and containing the vertex set are A326903.
%Y A326905 Set-systems closed under intersection are A326901, with unlabeled version A326904.
%Y A326905 Cf. A006058, A102895, A102898, A326866, A326876, A326878, A326882, A326900, A326902.
%K A326905 nonn
%O A326905 1,3
%A A326905 _Gus Wiseman_, Aug 04 2019