A327109 -id:A327109 - cp's OEIS frontend

A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).

4, 5, 6, 7, 16, 17, 24, 25, 32, 34, 40, 42, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107

Offset: 1

Gus Wiseman, Aug 22 2019

nonn

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.

A set-system is 2-cut-connected if any single vertex can be removed (along with any empty edges) without making the set-system disconnected or empty. Except for cointersecting set-systems (A326853), this is the same as 2-vertex-connectivity.

			The sequence of all 2-cut-connected set-systems together with their BII-numbers begins:
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  63: {{1},{2},{1,2},{3},{1,3},{2,3}}

Positions of numbers >= 2 in A326786.
2-cut-connected graphs are counted by A013922, if we assume A013922(2) = 0.
2-cut-connected integer partitions are counted by A322387.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for cut-connectivity 1 are A327098.
BII-numbers for non-spanning edge-connectivity >= 2 are A327102.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Covering 2-cut-connected set-systems are counted by A327112.
Covering set-systems with cut-connectivity 2 are counted by A327113.
The labeled cut-connectivity triangle is A327125, with unlabeled version A327127.
Cf. A000120, A002218, A048793, A070939, A259862, A322388, A326031, A326749, A327097, A327108.

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]]]]]]]]];
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],cutConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]>=2&]

If (*) is intersection and (-) is complement, we have A327101 * A326704 = A326751 - A058891, i.e., the intersection of A327101 (this sequence) with A326704 (antichains) is the complement of A058891 (singletons) in A326751 (blobs).

A327074 Number of unlabeled connected graphs with n vertices and exactly one bridge.

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 25, 197, 2454, 48201, 1604016, 93315450, 9696046452, 1822564897453, 625839625866540, 395787709599238772, 464137745175250610865, 1015091996575508453655611, 4160447945769725861550193834, 32088553211819016484736085677320, 467409605282347770524641700949750858

Offset: 0

Gus Wiseman, Aug 24 2019

nonn

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Unlabeled graphs with no bridges are counted by A007146 (unlabeled graphs with spanning edge-connectivity >= 2).

Gus Wiseman, The a(5) = 4 unlabeled connected graphs with n vertices and exactly one bridge.

The labeled version is A327073.
Unlabeled graphs with at least one bridge are A052446.
The enumeration of unlabeled connected graphs by number of bridges is A327077.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Cf. A001349, A002494, A007146, A263296, A327075, A327111, A327144, A327145.

A007145 = Cases[Import["https://oeis.org/A007145/b007145.txt", "Table"], {, }][[All, 2]];
f[x_] = A007145 . x^Range[lg = Length[A007145]];
(f[x]^2 + f[x^2])/2 + O[x]^lg // CoefficientList[#, x]& (* Jean-François Alcover, Sep 23 2019 *)

G.f.: (f(x)^2 + f(x^2))/2 where f(x) is the g.f. of A007145. - Andrew Howroyd, Aug 25 2019

Terms a(6) and beyond from Andrew Howroyd, Aug 25 2019

A327110 BII-numbers of set-systems with spanning edge-connectivity 3.

Original entry on oeis.org

116, 117, 118, 119, 124, 125, 126, 127, 1796, 1797, 1798, 1799, 1904, 1905, 1906, 1907, 1908, 1909, 1910, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1918, 1919, 1924, 1925, 1926, 1927, 2032, 2033, 2034, 2035, 2036, 2037, 2038, 2039, 2040, 2041, 2042, 2043, 2044

Offset: 1

Gus Wiseman, Oct 03 2019

nonn

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			The sequence of all set-systems with spanning edge-connectivity 3 together with their BII-numbers begins:
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   117: {{1},{1,2},{1,3},{2,3},{1,2,3}}
   118: {{2},{1,2},{1,3},{2,3},{1,2,3}}
   119: {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
   124: {{1,2},{3},{1,3},{2,3},{1,2,3}}
   125: {{1},{1,2},{3},{1,3},{2,3},{1,2,3}}
   126: {{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
   127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
  1796: {{1,2},{1,4},{2,4},{1,2,4}}
  1797: {{1},{1,2},{1,4},{2,4},{1,2,4}}
  1798: {{2},{1,2},{1,4},{2,4},{1,2,4}}
  1799: {{1},{2},{1,2},{1,4},{2,4},{1,2,4}}
  1904: {{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1905: {{1},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1906: {{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1907: {{1},{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1908: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1909: {{1},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1910: {{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1911: {{1},{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}

Positions of 3's in A327144.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A001187, A095983, A322395, A326749, A326753, A326787, A327069, A327071, A327103, A327130, A327146, A327147.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[1000],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==3&]

cp's OEIS Frontend

A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).

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A327074 Number of unlabeled connected graphs with n vertices and exactly one bridge.

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A327110 BII-numbers of set-systems with spanning edge-connectivity 3.

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