A322387 -id:A322387 - cp's OEIS frontend

A322395 Number of labeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

1, 1, 1, 4, 26, 548, 22504, 1708336, 241874928, 65285161232, 34305887955616, 35573982726480064, 73308270568902715136, 301210456065963448091072, 2471487759846321319412778624, 40526856087731237340916330352896, 1328570640536613080046570271722309632

Offset: 0

Gus Wiseman, Dec 06 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Bridge
Eric Weisstein's World of Mathematics, Endpoint

Cf. A001187, A006125, A007146, A013922, A054921, A095983, A322338, A322387, A322394.

nmax = 16;
seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n + 1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k - 1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];
A095983 = seq[nmax];
a[n_] := If[n<3, 1, n+Sum[Binomial[n, k]*A095983[[k+1]]*k^(n-k), {k, 1, n}]];
a /@ Range[0, nmax] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

a(n) = n + Sum_{k=1..n} binomial(n,k)*A095983(k)*k^(n-k) for n >= 3. - Andrew Howroyd, Dec 07 2018

a(6)-a(16) from Andrew Howroyd, Dec 07 2018

A322389 Vertex-connectivity of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Dec 05 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

The vertex-connectivity of an integer partition is the minimum number of primes that must be divided out (and any parts then equal to 1 removed) so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			The integer partition (6,4,3) with Heinz number 455 does not become disconnected or empty if 2 is divided out giving (3,3), or if 3 is divided out giving (4,2), but it does become disconnected or empty if both 2 and 3 are divided out giving (); so a(455) = 2.
195 is the Heinz number of (6,3,2), corresponding to the multiset partition {{1},{2},{1,2}}. Removing the vertex 1 gives {{2},{2}}, while removing 2 gives {{1},{1}}. These are both connected, so both vertices must be removed to obtain a disconnected or empty multiset partition; hence a(195) = 2.

Wikipedia, k-vertex-connected graph

Cf. A003963, A013922, A056239, A095983, A112798, A302242, A304716, A305078, A305079, A322335, A322336, A322338, A322387, A322388, A322390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Array[vertConn@*primeMS,100]

A052443 Number of simple unlabeled n-node graphs of connectivity 2.

Original entry on oeis.org

0, 0, 1, 2, 7, 39, 332, 4735, 113176, 4629463, 327695586, 40525166511, 8850388574939, 3453378695335727, 2435485662537561705, 3137225298932374490227, 7448146273273417700880931, 32837456713651735794742705141, 270528237651574516777595556494978, 4186091025846007046878947026003803389

Offset: 1

Eric W. Weisstein

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..23
Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Connected Graph.
Gus Wiseman, The a(5) = 7 graphs with vertex-connectivity 2.

Column k=2 of A259862.
Cf. A000719, A002218, A006290, A052442, A052444, A052445.
The labeled version is A327198.
2-vertex-connected graphs are A013922.
Cf. A322387, A327051, A327101, A327334.

A002218 = Cases[Import["https://oeis.org/A002218/b002218.txt", "Table"], {, }][[All, 2]];
A006290 = Cases[Import["https://oeis.org/A006290/b006290.txt", "Table"], {, }][[All, 2]];
a[1] = 0; a[2] = 0; a[3] = 1;
a[n_] := A002218[[n]] - A006290[[n-3]];
Array[a, 23] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

a(n) = A002218(n) - A006290(n) for n > 2. - Andrew Howroyd, Sep 04 2019

Name clarified and a(8)-a(11) by Jens M. Schmidt, Feb 18 2019
a(2)-a(3) corrected by Andrew Howroyd, Aug 28 2019
a(12)-a(20) from Andrew Howroyd, Sep 04 2019

A322390 Number of integer partitions of n with vertex-connectivity 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 8, 1, 7, 3, 11, 1, 14, 2, 18, 7, 21, 6, 35, 14, 43, 28, 65, 42, 96, 70, 141, 120, 205, 187, 315, 286, 445, 445, 657

Offset: 1

Gus Wiseman, Dec 05 2018

The vertex-connectivity of an integer partition is the minimum number of primes that must be divided out (and any parts then equal to 1 removed) so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			The a(14) = 7 integer partitions are (842), (8222), (77), (4442), (44222), (422222), (2222222).
The a(18) = 14 integer partitions:
  (9,9), (16,2),
  (8,8,2), (10,6,2),
  (8,4,4,2), (9,3,3,3),
  (4,4,4,4,2), (8,4,2,2,2),
  (3,3,3,3,3,3), (4,4,4,2,2,2), (8,2,2,2,2,2),
  (4,4,2,2,2,2,2),
  (4,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2).

Wikipedia, k-vertex-connected graph

Cf. A013922, A054921, A095983, A304714, A304716, A305078, A305079, A322335, A322338, A322387, A322389, A322391.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[IntegerPartitions[n],vertConn[#]==1&]],{n,20}]

A327082 BII-numbers of set-systems with cut-connectivity 2.

Original entry on oeis.org

4, 5, 6, 7, 16, 17, 24, 25, 32, 34, 40, 42, 256, 257, 384, 385, 512, 514, 640, 642, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850

Offset: 1

Gus Wiseman, Aug 20 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.

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).

			The sequence of all set-systems with cut-connectivity 2 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}}
  256: {{1,4}}
  257: {{1},{1,4}}
  384: {{4},{1,4}}
  385: {{1},{4},{1,4}}
  512: {{2,4}}
  514: {{2},{2,4}}
  640: {{4},{2,4}}
  642: {{2},{4},{2,4}}
The first term involving an edge of size 3 is 832: {{1,2,3},{1,4},{2,4}}.

Positions of 2's in A326786.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity 2 are A327108.
The cut-connectivity 1 version is A327098.
The cut-connectivity > 1 version is A327101.
Covering 2-cut-connected set-systems are counted by A327112.
Covering set-systems with cut-connectivity 2 are counted by A327113.
Cf. A000120, A002218, A013922, A048793, A259862, A322387, A322388, A326031, A327041, A327114.

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[#]]==2&]

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

Original entry on oeis.org

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).

A327112 Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems.

Original entry on oeis.org

0, 0, 4, 72, 29856

Offset: 0

Gus Wiseman, Aug 24 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The cut-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity (A327334, A327051).

			Non-isomorphic representatives of the a(3) = 72 set-systems:
  {{123}}
  {{3}{123}}
  {{23}{123}}
  {{2}{3}{123}}
  {{1}{23}{123}}
  {{3}{23}{123}}
  {{12}{13}{23}}
  {{13}{23}{123}}
  {{1}{2}{3}{123}}
  {{1}{3}{23}{123}}
  {{2}{3}{23}{123}}
  {{3}{12}{13}{23}}
  {{2}{13}{23}{123}}
  {{3}{13}{23}{123}}
  {{12}{13}{23}{123}}
  {{1}{2}{3}{23}{123}}
  {{2}{3}{12}{13}{23}}
  {{1}{2}{13}{23}{123}}
  {{2}{3}{13}{23}{123}}
  {{3}{12}{13}{23}{123}}
  {{1}{2}{3}{12}{13}{23}}
  {{1}{2}{3}{13}{23}{123}}
  {{2}{3}{12}{13}{23}{123}}
  {{1}{2}{3}{12}{13}{23}{123}}

Covering 2-cut-connected graphs are A013922, if we assume A013922(2) = 1.
Covering 1-cut-connected antichains (clutters) are A048143, if we assume A048143(0) = A048143(1) =0.
Covering 2-cut-connected antichains (blobs) are A275307, if we assume A275307(1) = 0.
Covering set-systems with cut-connectivity 2 are A327113.
2-vertex-connected integer partitions are A322387.
BII-numbers of set-systems with cut-connectivity >= 2 are A327101.
The cut-connectivity of the set-system with BII-number n is A326786(n).
Cf. A002218, A003465, A013922, A259862, A323818, A327082, A327126, A327130.

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]]]]]]]]];
vConn[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&vConn[#]>=2&]],{n,0,3}]

A327113 Number of set-systems covering n vertices with cut-connectivity 2.

Original entry on oeis.org

0, 0, 4, 0, 4752

Offset: 0

Gus Wiseman, Aug 24 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The cut-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity (A327334, A327051).

			The a(2) = 4 set-systems:
  {{1,2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Covering graphs with cut-connectivity >= 2 are A013922, if we assume A013922(2) = 1.
Covering antichains (blobs) with cut-connectivity >= 2 are A275307, if we assume A275307(1) = 0.
2-vertex-connected integer partitions are A322387.
Connected covering set-systems are A323818.
Covering set-systems with cut-connectivity >= 2 are A327112.
The cut-connectivity of the set-system with BII-number n is A326786(n).
BII-numbers of set-systems with cut-connectivity 2 are A327082.
Cf. A002218, A003465, A048143, A259862, A322389, A327101, A327113, A327126, A327128, A327130.

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]]]]]]]]];
vConn[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&vConn[#]==2&]],{n,0,3}]

A322388 Heinz numbers of 2-vertex-connected integer partitions.

Original entry on oeis.org

13, 29, 37, 39, 43, 47, 61, 65, 71, 73, 79, 87, 89, 91, 101, 107, 111, 113, 117, 129, 137, 139, 149, 151, 163, 167, 169, 173, 181, 183, 185, 193, 195, 197, 199, 203, 213, 223, 229, 233, 235, 237, 239, 247, 251, 257, 259, 261, 263, 267, 269, 271, 273, 281

Offset: 1

Gus Wiseman, Dec 05 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).

			The sequence of all 2-vertex-connected integer partitions begins: (1), (6), (10), (12), (6,2), (14), (15), (18), (6,3), (20), (21), (22), (10,2), (24), (6,4), (26), (28), (12,2), (30), (6,2,2), (14,2), (33), (34), (35), (36), (38), (39), (6,6), (40), (42), (18,2), (12,3), (44), (6,3,2), (45), (46).

Wikipedia, k-vertex-connected graph

Cf. A003963, A013922, A056239, A095983, A112798, A218970, A275307, A304716, A305078, A305079, A322336, A322338, A322387, A322389, A322390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]]
Select[Range[100],vertConn[primeMS[#]]>1&]

A322391 Number of integer partitions of n with edge-connectivity 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 9, 3, 14, 8, 17, 13, 35, 17, 49, 35, 67, 53, 114, 69

Offset: 1

Gus Wiseman, Dec 05 2018

The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			The a(20) = 8 integer partitions:
  (20),
  (12,3,3,2), (9,6,3,2), (8,6,3,3),
  (6,4,4,3,3),
  (6,4,3,3,2,2), (6,3,3,3,3,2),
  (6,3,3,2,2,2,2).

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A304716, A305078, A305079, A322335, A322336, A322387, A322390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]];
Table[Length[Select[IntegerPartitions[n],edgeConn[#]==1&]],{n,20}]

cp's OEIS Frontend

A322395 Number of labeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

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A322389 Vertex-connectivity of the integer partition with Heinz number n.

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A052443 Number of simple unlabeled n-node graphs of connectivity 2.

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A322390 Number of integer partitions of n with vertex-connectivity 1.

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A327082 BII-numbers of set-systems with cut-connectivity 2.

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Mathematica

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

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Formula

A327112 Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems.

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Mathematica

A327113 Number of set-systems covering n vertices with cut-connectivity 2.

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Mathematica

A322388 Heinz numbers of 2-vertex-connected integer partitions.

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