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A327098 BII-numbers of set-systems with cut-connectivity 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 (list; graph; refs; listen; history; text; internal format)
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 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.

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).

LINKS

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

EXAMPLE

The sequence of all set-systems with cut-connectivity 1 together with their BII-numbers 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.

BII-numbers for cut-connectivity 2 are A327082.

BII-numbers for non-spanning edge-connectivity 1 are A327099.

BII-numbers for spanning edge-connectivity 1 are A327111.

Integer partitions with cut-connectivity 1 are counted by A322390.

Labeled connected separable graphs are counted by A327114.

Connected separable set-systems 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

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Last modified March 3 09:46 EST 2021. Contains 341760 sequences. (Running on oeis4.)