cp's OEIS Frontend

A327099 BII-numbers of set-systems with non-spanning edge-connectivity 1.

%I A327099 #5 Aug 22 2019 20:41:43
%S A327099 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 A327099 57,58,59,64,67,73,74,75,76,77,78,79,82,83,90,91,97,99,105,107,128,
%U A327099 256,262,263,278,279,280,281,284,285,286,287,292,293,294,295,300
%N A327099 BII-numbers of set-systems with non-spanning edge-connectivity 1.
%C A327099 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 A327099 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 A327099 The sequence of all set-systems with non-spanning edge-connectivity 1 together with their BII-numbers begins:
%e A327099    1: {{1}}
%e A327099    2: {{2}}
%e A327099    4: {{1,2}}
%e A327099    7: {{1},{2},{1,2}}
%e A327099    8: {{3}}
%e A327099   16: {{1,3}}
%e A327099   22: {{2},{1,2},{1,3}}
%e A327099   23: {{1},{2},{1,2},{1,3}}
%e A327099   25: {{1},{3},{1,3}}
%e A327099   28: {{1,2},{3},{1,3}}
%e A327099   29: {{1},{1,2},{3},{1,3}}
%e A327099   30: {{2},{1,2},{3},{1,3}}
%e A327099   31: {{1},{2},{1,2},{3},{1,3}}
%e A327099   32: {{2,3}}
%e A327099   37: {{1},{1,2},{2,3}}
%e A327099   39: {{1},{2},{1,2},{2,3}}
%e A327099   42: {{2},{3},{2,3}}
%e A327099   44: {{1,2},{3},{2,3}}
%e A327099   45: {{1},{1,2},{3},{2,3}}
%e A327099   46: {{2},{1,2},{3},{2,3}}
%t A327099 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327099 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 A327099 edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
%t A327099 Select[Range[0,100],edgeConn[bpe[#]]==1&]
%Y A327099 Positions of 1's in A326787.
%Y A327099 Simple graphs with non-spanning edge-connectivity 1 are A327071.
%Y A327099 BII-numbers for non-spanning edge-connectivity >= 1 are A326749.
%Y A327099 BII-numbers for non-spanning edge-connectivity 2 are A327097.
%Y A327099 BII-numbers for spanning edge-connectivity 1 are A327111.
%Y A327099 BII-numbers for vertex-connectivity 1 are A327114.
%Y A327099 Covering set-systems with non-spanning edge-connectivity 1 are counted by A327129.
%Y A327099 Cf. A048793, A052446, A070939, A322338, A326031, A327108.
%K A327099 nonn
%O A327099 1,2
%A A327099 _Gus Wiseman_, Aug 21 2019