A322390 -id:A322390 - cp's OEIS frontend

A326786 Cut-connectivity of the set-system with BII-number n.

0, 1, 1, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 1, 1, 1, 2, 2, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Gus Wiseman, Jul 25 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 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. The cut-connectivity of a set-system is the minimum number of vertices that must be removed (together with any resulting empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A326853), this is the same as vertex-connectivity (A327051).

			Positions of first appearances of each integer, together with the corresponding set-systems, are:
     0: {}
     1: {{1}}
     4: {{1,2}}
    52: {{1,2},{1,3},{2,3}}
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

Cf. A000120, A013922, A029931, A048793, A070939, A305078, A322388, A322389 (same for MM-numbers), A322390, A326031, A326701, A326749, A326753, A326787 (edge-connectivity), A327051 (vertex-connectivity).

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]]]]]]]]];
vertConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@bpe/@y],Function[del,Length[csm[DeleteCases[DeleteCases[bpe/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[vertConn[bpe[n]],{n,0,100}]

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]

A327076 Maximum divisor of n that is 1 or connected.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 9, 5, 11, 3, 13, 7, 5, 2, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 2, 11, 17, 7, 9, 37, 19, 39, 5, 41, 21, 43, 11, 9, 23, 47, 3, 49, 25, 17, 13, 53, 27, 11, 7, 57, 29, 59, 5, 61, 31, 63, 2, 65, 11, 67, 17, 23, 7, 71, 9, 73

Offset: 1

Gus Wiseman, Sep 05 2019

nonn

A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_k is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078, which is the union of this sequence without 1.

Also the maximum MM-number (A302242) of a connected subset of the multiset of multisets with MM-number n.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Positions of prime numbers are A302569.
Connected numbers are A305078.
Cf. A007947, A056239, A112798, A286518, A302242, A304716, A305079, A322338, A322390, A322391.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Max[Select[Divisors[n],Length[zsm[primeMS[#]]]<=1&]],{n,30}]

If n is in A305078, then a(n) = n.

A327098 BII-numbers of set-systems with cut-connectivity 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 21 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 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}}

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.

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&]

A327114 Number of labeled simple graphs covering n vertices with cut-connectivity 1.

Original entry on oeis.org

0, 0, 0, 3, 28, 490, 15336, 851368, 85010976, 15615858960, 5388679220480, 3548130389657216, 4507988483733389568, 11145255551131555572992, 53964198507018134569758720, 514158235191699333805861463040

Offset: 0

Gus Wiseman, Aug 25 2019

nonn

The cut-connectivity of a graph is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a disconnected or empty graph.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k = 1 of A327126.
The unlabeled version is A052442, if we assume A052442(2) = 0.
Connected non-separable graphs are A013922.
BII-numbers for cut-connectivity 1 are A327098.
Set-systems with cut-connectivity 1 are counted by A327197.
Labeled simple graphs with vertex-connectivity 1 are A327336.
Cf. A001187, A054592, A259862, A322389, A322390, A326786, A327070, A327100, A327125.

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]]]]]]]]];
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]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==1&]],{n,0,3}]

seq(n)={my(g=log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))); Vec(serlaplace(g-intformal(1+log(x/serreverse(x*deriv(g))))), -(n+1))} \\ Andrew Howroyd, Sep 11 2019

a(n) = A001187(n) - A013922(n), if we assume A001187(1) = 0.

A322387 Number of 2-vertex-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 1, 6, 2, 10, 8, 13, 9, 26, 14, 35, 28, 50, 37, 77, 54, 101, 84, 138, 110, 205, 149, 252, 222, 335, 287, 455, 375, 577, 522, 740, 657, 985

Offset: 1

Gus Wiseman, Dec 05 2018

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 a(14) = 10 2-vertex-connected integer partitions:
  (14)  (8,6)   (6,4,4)   (6,3,3,2)  (6,2,2,2,2)
        (10,4)  (6,6,2)   (6,4,2,2)
        (12,2)  (10,2,2)

Wikipedia, k-vertex-connected graph

Cf. A013922, A095983, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322388, 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]]];
Table[Length[Select[IntegerPartitions[n],vertConn[#]>1&]],{n,30}]

a(41)-a(42) from Jinyuan Wang, Jun 20 2020

A327336 Number of labeled simple graphs with vertex-connectivity 1.

Original entry on oeis.org

0, 0, 1, 3, 28, 490, 15336, 851368, 85010976, 15615858960, 5388679220480, 3548130389657216, 4507988483733389568, 11145255551131555572992, 53964198507018134569758720, 514158235191699333805861463040, 9672967865350359173180572164444160

Offset: 0

Gus Wiseman, Sep 02 2019

nonn

Same as A327114 except a(2) = 1.

The vertex-connectivity of a graph is the minimum number of vertices that must be removed (along with any incident edges) to obtain a non-connected graph or singleton.

			The a(2) = 1 through a(4) = 28 edge-sets:
  {12}  {12,13}  {12,13,14}
        {12,23}  {12,13,24}
        {13,23}  {12,13,34}
                 {12,14,23}
                 {12,14,34}
                 {12,23,24}
                 {12,23,34}
                 {12,24,34}
                 {13,14,23}
                 {13,14,24}
                 {13,23,24}
                 {13,23,34}
                 {13,24,34}
                 {14,23,24}
                 {14,23,34}
                 {14,24,34}
                 {12,13,14,23}
                 {12,13,14,24}
                 {12,13,14,34}
                 {12,13,23,24}
                 {12,13,23,34}
                 {12,14,23,24}
                 {12,14,24,34}
                 {12,23,24,34}
                 {13,14,23,34}
                 {13,14,24,34}
                 {13,23,24,34}
                 {14,23,24,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k = 1 of A327334.
The unlabeled version is A052442.
Connected non-separable graphs are A013922.
Set-systems with vertex-connectivity 1 are A327128.
Labeled simple graphs with cut-connectivity 1 are A327114.
Cf. A006129, A054592, A322389, A322390, A326786, A327070, A327098, A327100, A327125, A327126.

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]]]]]]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],vertConnSys[Range[n],#]==1&]],{n,0,4}]

Terms a(6) and beyond from Andrew Howroyd, Sep 11 2019

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&]

A327100 BII-numbers of antichains of sets with cut-connectivity 1.

Original entry on oeis.org

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

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.

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).

			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}}

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.

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&]

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

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

A326786 Cut-connectivity of the set-system with BII-number n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A327076 Maximum divisor of n that is 1 or connected.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A327098 BII-numbers of set-systems with cut-connectivity 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327114 Number of labeled simple graphs covering n vertices with cut-connectivity 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A322387 Number of 2-vertex-connected integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A327336 Number of labeled simple graphs with vertex-connectivity 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs