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 A327109 BII-numbers of set-systems with spanning edge-connectivity >= 2. 13

%I

%S 52,53,54,55,60,61,62,63,84,85,86,87,92,93,94,95,100,101,102,103,108,

%T 109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,

%U 126,127,772,773,774,775,816,817,818,819,820,821,822,823,824,825,826

%N BII-numbers of set-systems with spanning edge-connectivity >= 2.

%C Differs from A327108 in having 116, 117, 118, 119, 124, 125, 126, 127, ...

%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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

%e The sequence of all set-systems with spanning edge-connectivity >= 2 together with their BII-numbers begins:

%e 52: {{1,2},{1,3},{2,3}}

%e 53: {{1},{1,2},{1,3},{2,3}}

%e 54: {{2},{1,2},{1,3},{2,3}}

%e 55: {{1},{2},{1,2},{1,3},{2,3}}

%e 60: {{1,2},{3},{1,3},{2,3}}

%e 61: {{1},{1,2},{3},{1,3},{2,3}}

%e 62: {{2},{1,2},{3},{1,3},{2,3}}

%e 63: {{1},{2},{1,2},{3},{1,3},{2,3}}

%e 84: {{1,2},{1,3},{1,2,3}}

%e 85: {{1},{1,2},{1,3},{1,2,3}}

%e 86: {{2},{1,2},{1,3},{1,2,3}}

%e 87: {{1},{2},{1,2},{1,3},{1,2,3}}

%e 92: {{1,2},{3},{1,3},{1,2,3}}

%e 93: {{1},{1,2},{3},{1,3},{1,2,3}}

%e 94: {{2},{1,2},{3},{1,3},{1,2,3}}

%e 95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}

%e 100: {{1,2},{2,3},{1,2,3}}

%e 101: {{1},{1,2},{2,3},{1,2,3}}

%e 102: {{2},{1,2},{2,3},{1,2,3}}

%e 103: {{1},{2},{1,2},{2,3},{1,2,3}}

%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 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];

%t Select[Range[0,1000],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]>=2&]

%Y Positions of terms >= 2 in A327144.

%Y Graphs with spanning edge-connectivity >= 2 are counted by A095983.

%Y Graphs with spanning edge-connectivity 2 are counted by A327146.

%Y Set-systems with spanning edge-connectivity 2 are counted by A327130.

%Y BII-numbers for non-spanning edge-connectivity 2 are A327097.

%Y BII-numbers for non-spanning edge-connectivity >= 2 are A327102.

%Y BII-numbers for spanning edge-connectivity 2 are A327108.

%Y BII-numbers for spanning edge-connectivity 1 are A327111.

%Y Cf. A326749, A326753, A326787, A327041, A327069, A327071, A327075.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 23 2019

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Last modified April 11 22:01 EDT 2021. Contains 342888 sequences. (Running on oeis4.)