|
%I #15 Sep 01 2019 22:02:54
%S 4,5,6,7,16,17,24,25,32,34,40,42,256,257,384,385,512,514,640,642,816,
%T 817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,
%U 834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850
%N BII-numbers of set-systems with cut-connectivity 2.
%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 We define the cut-connectivity (A326786, A327237), of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex and no edges has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039), this is the same as vertex-connectivity (A327334, A327051).
%e The sequence of all set-systems with cut-connectivity 2 together with their BII-numbers begins:
%e 4: {{1,2}}
%e 5: {{1},{1,2}}
%e 6: {{2},{1,2}}
%e 7: {{1},{2},{1,2}}
%e 16: {{1,3}}
%e 17: {{1},{1,3}}
%e 24: {{3},{1,3}}
%e 25: {{1},{3},{1,3}}
%e 32: {{2,3}}
%e 34: {{2},{2,3}}
%e 40: {{3},{2,3}}
%e 42: {{2},{3},{2,3}}
%e 256: {{1,4}}
%e 257: {{1},{1,4}}
%e 384: {{4},{1,4}}
%e 385: {{1},{4},{1,4}}
%e 512: {{2,4}}
%e 514: {{2},{2,4}}
%e 640: {{4},{2,4}}
%e 642: {{2},{4},{2,4}}
%e The first term involving an edge of size 3 is 832: {{1,2,3},{1,4},{2,4}}.
%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 vertConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
%t Select[Range[0,100],vertConnSys[bpe/@bpe[#]]==2&]
%Y Positions of 2's in A326786.
%Y BII-numbers for non-spanning edge-connectivity 2 are A327097.
%Y BII-numbers for spanning edge-connectivity 2 are A327108.
%Y The cut-connectivity 1 version is A327098.
%Y The cut-connectivity > 1 version is A327101.
%Y Covering 2-cut-connected set-systems are counted by A327112.
%Y Covering set-systems with cut-connectivity 2 are counted by A327113.
%Y Cf. A000120, A002218, A013922, A048793, A259862, A322387, A322388, A326031, A327041, A327114.
%K nonn
%O 1,1
%A _Gus Wiseman_, Aug 20 2019
|
