

A327100


BIInumbers of antichains of sets with cutconnectivity 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
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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 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 has cutconnectivity 1. Except for cointersecting setsystems (A326853, A327039, A327040), this is the same as vertexconnectivity (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 vertexconnectivity 1 together with their BIInumbers 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 vertexconnectivity >= 1 are A326750.
BIInumbers for cutconnectivity 2 are A327082.
BIInumbers for cutconnectivity 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



