cp's OEIS Frontend

A327111 BII-numbers of set-systems with spanning edge-connectivity 1.

%I A327111 #6 Aug 26 2019 12:40:08
%S A327111 1,2,4,5,6,7,8,16,17,20,21,22,23,24,25,28,29,30,31,32,34,36,37,38,39,
%T A327111 40,42,44,45,46,47,48,49,50,51,56,57,58,59,64,65,66,67,68,69,70,71,72,
%U A327111 73,74,75,76,77,78,79,80,81,82,83,88,89,90,91,96,97,98,99
%N A327111 BII-numbers of set-systems with spanning edge-connectivity 1.
%C A327111 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 A327111 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 A327111 The sequence of all set-systems with spanning edge-connectivity 1 together with their BII-numbers begins:
%e A327111    1: {{1}}
%e A327111    2: {{2}}
%e A327111    4: {{1,2}}
%e A327111    5: {{1},{1,2}}
%e A327111    6: {{2},{1,2}}
%e A327111    7: {{1},{2},{1,2}}
%e A327111    8: {{3}}
%e A327111   16: {{1,3}}
%e A327111   17: {{1},{1,3}}
%e A327111   20: {{1,2},{1,3}}
%e A327111   21: {{1},{1,2},{1,3}}
%e A327111   22: {{2},{1,2},{1,3}}
%e A327111   23: {{1},{2},{1,2},{1,3}}
%e A327111   24: {{3},{1,3}}
%e A327111   25: {{1},{3},{1,3}}
%e A327111   28: {{1,2},{3},{1,3}}
%e A327111   29: {{1},{1,2},{3},{1,3}}
%e A327111   30: {{2},{1,2},{3},{1,3}}
%e A327111   31: {{1},{2},{1,2},{3},{1,3}}
%e A327111   32: {{2,3}}
%t A327111 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327111 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 A327111 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t A327111 Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]
%Y A327111 Graphs with spanning edge-connectivity >= 2 are counted by A095983.
%Y A327111 BII-numbers for vertex-connectivity 1 are A327098.
%Y A327111 BII-numbers for non-spanning edge-connectivity 1 are A327099.
%Y A327111 BII-numbers for spanning edge-connectivity 2 are A327108.
%Y A327111 BII-numbers for spanning edge-connectivity >= 2 are A327109.
%Y A327111 Set-systems with spanning edge-connectivity 2 are counted by A327130.
%Y A327111 Graphs with spanning edge-connectivity 1 are counted by A327145.
%Y A327111 Graphs with spanning edge-connectivity 2 are counted by A327146.
%Y A327111 Cf. A013922, A322395, A326749, A327041, A327069, A327071, A327097, A327144, A327145.
%K A327111 nonn
%O A327111 1,2
%A A327111 _Gus Wiseman_, Aug 25 2019