cp's OEIS Frontend

A327098 BII-numbers of set-systems with cut-connectivity 1.

%I A327098 #15 Sep 01 2019 22:03:07
%S A327098 1,2,8,20,21,22,23,28,29,30,31,36,37,38,39,44,45,46,47,48,49,50,51,56,
%T A327098 57,58,59,128,260,261,262,263,272,273,276,277,278,279,280,281,284,285,
%U A327098 286,287,292,293,294,295,300,301,302,303,304,305,306,307,308,309
%N A327098 BII-numbers of set-systems with cut-connectivity 1.
%C A327098 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 A327098 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 A327098 The sequence of all set-systems with cut-connectivity 1 together with their BII-numbers begins:
%e A327098    1: {{1}}
%e A327098    2: {{2}}
%e A327098    8: {{3}}
%e A327098   20: {{1,2},{1,3}}
%e A327098   21: {{1},{1,2},{1,3}}
%e A327098   22: {{2},{1,2},{1,3}}
%e A327098   23: {{1},{2},{1,2},{1,3}}
%e A327098   28: {{1,2},{3},{1,3}}
%e A327098   29: {{1},{1,2},{3},{1,3}}
%e A327098   30: {{2},{1,2},{3},{1,3}}
%e A327098   31: {{1},{2},{1,2},{3},{1,3}}
%e A327098   36: {{1,2},{2,3}}
%e A327098   37: {{1},{1,2},{2,3}}
%e A327098   38: {{2},{1,2},{2,3}}
%e A327098   39: {{1},{2},{1,2},{2,3}}
%e A327098   44: {{1,2},{3},{2,3}}
%e A327098   45: {{1},{1,2},{3},{2,3}}
%e A327098   46: {{2},{1,2},{3},{2,3}}
%e A327098   47: {{1},{2},{1,2},{3},{2,3}}
%e A327098   48: {{1,3},{2,3}}
%t A327098 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327098 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 A327098 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 A327098 Select[Range[0,100],vertConnSys[bpe/@bpe[#]]==1&]
%Y A327098 A subset of A326749.
%Y A327098 Positions of 1's in A326786.
%Y A327098 BII-numbers for cut-connectivity 2 are A327082.
%Y A327098 BII-numbers for non-spanning edge-connectivity 1 are A327099.
%Y A327098 BII-numbers for spanning edge-connectivity 1 are A327111.
%Y A327098 Integer partitions with cut-connectivity 1 are counted by A322390.
%Y A327098 Labeled connected separable graphs are counted by A327114.
%Y A327098 Connected separable set-systems are counted by A327197.
%Y A327098 Cf. A000120, A048793, A070939, A322389, A326031, A327100, A327125.
%K A327098 nonn
%O A327098 1,2
%A A327098 _Gus Wiseman_, Aug 21 2019