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%I #5 Aug 22 2019 20:41:43
%S 1,2,4,7,8,16,22,23,25,28,29,30,31,32,37,39,42,44,45,46,47,49,50,51,
%T 57,58,59,64,67,73,74,75,76,77,78,79,82,83,90,91,97,99,105,107,128,
%U 256,262,263,278,279,280,281,284,285,286,287,292,293,294,295,300
%N BII-numbers of set-systems with non-spanning edge-connectivity 1.
%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 set-system (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 non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to result in a disconnected or empty set-system.
%e The sequence of all set-systems with non-spanning edge-connectivity 1 together with their BII-numbers begins:
%e 1: {{1}}
%e 2: {{2}}
%e 4: {{1,2}}
%e 7: {{1},{2},{1,2}}
%e 8: {{3}}
%e 16: {{1,3}}
%e 22: {{2},{1,2},{1,3}}
%e 23: {{1},{2},{1,2},{1,3}}
%e 25: {{1},{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}}
%e 31: {{1},{2},{1,2},{3},{1,3}}
%e 32: {{2,3}}
%e 37: {{1},{1,2},{2,3}}
%e 39: {{1},{2},{1,2},{2,3}}
%e 42: {{2},{3},{2,3}}
%e 44: {{1,2},{3},{2,3}}
%e 45: {{1},{1,2},{3},{2,3}}
%e 46: {{2},{1,2},{3},{2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
%t Select[Range[0,100],edgeConn[bpe[#]]==1&]
%Y Positions of 1's in A326787.
%Y Simple graphs with non-spanning edge-connectivity 1 are A327071.
%Y BII-numbers for non-spanning edge-connectivity >= 1 are A326749.
%Y BII-numbers for non-spanning edge-connectivity 2 are A327097.
%Y BII-numbers for spanning edge-connectivity 1 are A327111.
%Y BII-numbers for vertex-connectivity 1 are A327114.
%Y Covering set-systems with non-spanning edge-connectivity 1 are counted by A327129.
%Y Cf. A048793, A052446, A070939, A322338, A326031, A327108.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 21 2019
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