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%I #7 Jul 31 2019 08:10:38
%S 0,4,16,32,64,68,80,84,96,100,112,116,256,288,512,528,1024,1028,1280,
%T 1284,1536,1540,1792,1796,2048,2052,4096,4112,4352,4368,6144,6160,
%U 6400,6416,8192,8224,8704,8736,10240,10272,10752,10784,16384,16388,16400,16416
%N BII-numbers of connectedness systems without singletons.
%C We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.
%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.
%C The enumeration of these set-systems by number of covered vertices is given by A326877.
%H Gus Wiseman, <a href="http://www.mathematica-journal.com/2017/12/every-clutter-is-a-tree-of-blobs/">Every Clutter Is a Tree of Blobs</a>, The Mathematica Journal, Vol. 19, 2017.
%e The sequence of all connectedness systems without singletons together with their BII-numbers begins:
%e 0: {}
%e 4: {{1,2}}
%e 16: {{1,3}}
%e 32: {{2,3}}
%e 64: {{1,2,3}}
%e 68: {{1,2},{1,2,3}}
%e 80: {{1,3},{1,2,3}}
%e 84: {{1,2},{1,3},{1,2,3}}
%e 96: {{2,3},{1,2,3}}
%e 100: {{1,2},{2,3},{1,2,3}}
%e 112: {{1,3},{2,3},{1,2,3}}
%e 116: {{1,2},{1,3},{2,3},{1,2,3}}
%e 256: {{1,4}}
%e 288: {{2,3},{1,4}}
%e 512: {{2,4}}
%e 528: {{1,3},{2,4}}
%e 1024: {{1,2,4}}
%e 1028: {{1,2},{1,2,4}}
%e 1280: {{1,4},{1,2,4}}
%e 1284: {{1,2},{1,4},{1,2,4}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t connnosQ[eds_]:=!MemberQ[Length/@eds,1]&&SubsetQ[eds,Union@@@Select[Tuples[eds,2],Intersection@@#!={}&]];
%t Select[Range[0,1000],connnosQ[bpe/@bpe[#]]&]
%Y Connectedness systems without singletons are counted by A072446, with unlabeled case A072444.
%Y Connectedness systems are counted by A326866, with unlabeled case A326867.
%Y BII-numbers of connectedness systems are A326872.
%Y The connected case is A326879.
%Y Cf. A029931, A048793, A072447, A326031, A326749, A326750, A326870, A326875, A326877.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jul 29 2019
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