cp's OEIS Frontend

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

%I A327109 #5 Aug 23 2019 14:15:56
%S A327109 52,53,54,55,60,61,62,63,84,85,86,87,92,93,94,95,100,101,102,103,108,
%T A327109 109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,
%U A327109 126,127,772,773,774,775,816,817,818,819,820,821,822,823,824,825,826
%N A327109 BII-numbers of set-systems with spanning edge-connectivity >= 2.
%C A327109 Differs from A327108 in having 116, 117, 118, 119, 124, 125, 126, 127, ...
%C A327109 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 A327109 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 A327109 The sequence of all set-systems with spanning edge-connectivity >= 2 together with their BII-numbers begins:
%e A327109    52: {{1,2},{1,3},{2,3}}
%e A327109    53: {{1},{1,2},{1,3},{2,3}}
%e A327109    54: {{2},{1,2},{1,3},{2,3}}
%e A327109    55: {{1},{2},{1,2},{1,3},{2,3}}
%e A327109    60: {{1,2},{3},{1,3},{2,3}}
%e A327109    61: {{1},{1,2},{3},{1,3},{2,3}}
%e A327109    62: {{2},{1,2},{3},{1,3},{2,3}}
%e A327109    63: {{1},{2},{1,2},{3},{1,3},{2,3}}
%e A327109    84: {{1,2},{1,3},{1,2,3}}
%e A327109    85: {{1},{1,2},{1,3},{1,2,3}}
%e A327109    86: {{2},{1,2},{1,3},{1,2,3}}
%e A327109    87: {{1},{2},{1,2},{1,3},{1,2,3}}
%e A327109    92: {{1,2},{3},{1,3},{1,2,3}}
%e A327109    93: {{1},{1,2},{3},{1,3},{1,2,3}}
%e A327109    94: {{2},{1,2},{3},{1,3},{1,2,3}}
%e A327109    95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
%e A327109   100: {{1,2},{2,3},{1,2,3}}
%e A327109   101: {{1},{1,2},{2,3},{1,2,3}}
%e A327109   102: {{2},{1,2},{2,3},{1,2,3}}
%e A327109   103: {{1},{2},{1,2},{2,3},{1,2,3}}
%t A327109 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327109 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 A327109 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t A327109 Select[Range[0,1000],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]>=2&]
%Y A327109 Positions of terms >= 2 in  A327144.
%Y A327109 Graphs with spanning edge-connectivity >= 2 are counted by A095983.
%Y A327109 Graphs with spanning edge-connectivity 2 are counted by A327146.
%Y A327109 Set-systems with spanning edge-connectivity 2 are counted by A327130.
%Y A327109 BII-numbers for non-spanning edge-connectivity 2 are A327097.
%Y A327109 BII-numbers for non-spanning edge-connectivity >= 2 are A327102.
%Y A327109 BII-numbers for spanning edge-connectivity 2 are A327108.
%Y A327109 BII-numbers for spanning edge-connectivity 1 are A327111.
%Y A327109 Cf. A326749, A326753, A326787, A327041, A327069, A327071, A327075.
%K A327109 nonn
%O A327109 1,1
%A A327109 _Gus Wiseman_, Aug 23 2019