

A327098


BIInumbers of setsystems with cutconnectivity 1.


15



1, 2, 8, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 128, 260, 261, 262, 263, 272, 273, 276, 277, 278, 279, 280, 281, 284, 285, 286, 287, 292, 293, 294, 295, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309
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OFFSET

1,2


COMMENTS

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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem (finite set of finite nonempty sets) has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.
We define the cutconnectivity (A326786, A327237), of a setsystem to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty setsystem, with the exception that a setsystem with one vertex and no edges has cutconnectivity 1. Except for cointersecting setsystems (A326853, A327039), this is the same as vertexconnectivity (A327334, A327051).


LINKS

Table of n, a(n) for n=1..58.


EXAMPLE

The sequence of all setsystems with cutconnectivity 1 together with their BIInumbers begins:
1: {{1}}
2: {{2}}
8: {{3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
22: {{2},{1,2},{1,3}}
23: {{1},{2},{1,2},{1,3}}
28: {{1,2},{3},{1,3}}
29: {{1},{1,2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
31: {{1},{2},{1,2},{3},{1,3}}
36: {{1,2},{2,3}}
37: {{1},{1,2},{2,3}}
38: {{2},{1,2},{2,3}}
39: {{1},{2},{1,2},{2,3}}
44: {{1,2},{3},{2,3}}
45: {{1},{1,2},{3},{2,3}}
46: {{2},{1,2},{3},{2,3}}
47: {{1},{2},{1,2},{3},{2,3}}
48: {{1,3},{2,3}}


MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
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]]]]]]]]];
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]]];
Select[Range[0, 100], vertConnSys[bpe/@bpe[#]]==1&]


CROSSREFS

A subset of A326749.
Positions of 1's in A326786.
BIInumbers for cutconnectivity 2 are A327082.
BIInumbers for nonspanning edgeconnectivity 1 are A327099.
BIInumbers for spanning edgeconnectivity 1 are A327111.
Integer partitions with cutconnectivity 1 are counted by A322390.
Labeled connected separable graphs are counted by A327114.
Connected separable setsystems are counted by A327197.
Cf. A000120, A048793, A070939, A322389, A326031, A327100, A327125.
Sequence in context: A188893 A227127 A227399 * A030097 A136904 A043002
Adjacent sequences: A327095 A327096 A327097 * A327099 A327100 A327101


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 21 2019


STATUS

approved



