login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A327100 BII-numbers of antichains of sets with cut-connectivity 1. 6
1, 2, 8, 20, 36, 48, 128, 260, 272, 276, 292, 304, 308, 320, 516, 532, 544, 548, 560, 564, 576, 768, 784, 788, 800, 804, 1040, 1056, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304, 2308, 2324, 2336, 2352, 2560, 2564, 2576, 2596, 2608, 2816, 2820, 2832, 2848 (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 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 has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).

LINKS

Table of n, a(n) for n=1..49.

FORMULA

If (+) is union and (-) is complement, we have A327100 = A058891 + (A326750 - A326751).

EXAMPLE

The sequence of all antichains of sets with vertex-connectivity 1 together with their BII-numbers begins:

    1: {{1}}

    2: {{2}}

    8: {{3}}

   20: {{1,2},{1,3}}

   36: {{1,2},{2,3}}

   48: {{1,3},{2,3}}

  128: {{4}}

  260: {{1,2},{1,4}}

  272: {{1,3},{1,4}}

  276: {{1,2},{1,3},{1,4}}

  292: {{1,2},{2,3},{1,4}}

  304: {{1,3},{2,3},{1,4}}

  308: {{1,2},{1,3},{2,3},{1,4}}

  320: {{1,2,3},{1,4}}

  516: {{1,2},{2,4}}

  532: {{1,2},{1,3},{2,4}}

  544: {{2,3},{2,4}}

  548: {{1,2},{2,3},{2,4}}

  560: {{1,3},{2,3},{2,4}}

  564: {{1,2},{1,3},{2,3},{2,4}}

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]]]]]]]]];

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];

cutConnSys[vts_, eds_]:=If[Length[vts]==1, 1, Min@@Length/@Select[Subsets[vts], Function[del, csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]]];

Select[Range[0, 100], stableQ[bpe/@bpe[#], SubsetQ]&&cutConnSys[Union@@bpe/@bpe[#], bpe/@bpe[#]]==1&]

CROSSREFS

Positions of 1's in A326786.

The graphical case is A327114.

BII numbers of antichains with vertex-connectivity >= 1 are A326750.

BII-numbers for cut-connectivity 2 are A327082.

BII-numbers for cut-connectivity 1 are A327098.

Cf. A000120, A000372, A006126, A048143, A048793, A070939, A322390, A326031, A326749, A326751, A327071, A327111.

Sequence in context: A217513 A071386 A031114 * A130238 A038460 A077588

Adjacent sequences:  A327097 A327098 A327099 * A327101 A327102 A327103

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 22 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 25 16:03 EDT 2021. Contains 346291 sequences. (Running on oeis4.)