

A326786


Cutconnectivity of the setsystem with BIInumber n.


30



0, 1, 1, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 1, 1, 1, 2, 2, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
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OFFSET

0,5


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 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. The cutconnectivity of a setsystem is the minimum number of vertices that must be removed (together with any resulting empty or duplicate edges) to obtain a disconnected or empty setsystem. Except for cointersecting setsystems (A326853), this is the same as vertexconnectivity (A327051).


LINKS

Table of n, a(n) for n=0..86.


EXAMPLE

Positions of first appearances of each integer, together with the corresponding setsystems, are:
0: {}
1: {{1}}
4: {{1,2}}
52: {{1,2},{1,3},{2,3}}
2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}


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]]]]]]]]];
vertConn[y_]:=If[Length[csm[bpe/@y]]!=1, 0, Min@@Length/@Select[Subsets[Union@@bpe/@y], Function[del, Length[csm[DeleteCases[DeleteCases[bpe/@y, Alternatives@@del, {2}], {}]]]!=1]]];
Table[vertConn[bpe[n]], {n, 0, 100}]


CROSSREFS

Cf. A000120, A013922, A029931, A048793, A070939, A305078, A322388, A322389 (same for MMnumbers), A322390, A326031, A326701, A326749, A326753, A326787 (edgeconnectivity), A327051 (vertexconnectivity).
Sequence in context: A268242 A309509 A216953 * A276206 A334222 A124752
Adjacent sequences: A326783 A326784 A326785 * A326787 A326788 A326789


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jul 25 2019


STATUS

approved



