A013922 -id:A013922 - cp's OEIS frontend

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

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&]

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

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).

A004115 Number of unlabeled rooted nonseparable graphs with n nodes.

Original entry on oeis.org

0, 1, 1, 4, 22, 178, 2278, 46380, 1578060, 92765486, 9676866173, 1821391854302, 625710416245358, 395761853562201960, 464128290507379386872, 1015085639712281997464676, 4160440039279630394986003604, 32088534920274236421098827156776

Offset: 1

N. J. A. Sloane

R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..40 (terms 1..26 from R. W. Robinson)
R. W. Robinson, Rooted non-separable graphs - computer printout

Cf. A002218, A007145, A013922.

\\ See links in A339645 for combinatorial species functions.
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!}
graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)}
cycleIndexSeries(n)={my(g=graphsSeries(n), gcr=sPoint(g)/g); x*sSolve( sLog( gcr/(x*sv(1)) ), gcr )}
{ my(N=15); Vec(OgfSeries(cycleIndexSeries(N)), -N) } \\ Andrew Howroyd, Dec 25 2020

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}]

A317674 Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 84, 23, 6, 1, 6348, 470, 65, 10, 1, 7743728, 39598, 1575, 145, 15, 1, 2414572893530, 54354104, 144403, 4095, 280, 21, 1, 56130437190053299918162, 19316801997024, 218033088, 402073, 9100, 490, 28, 1

Offset: 1

Gus Wiseman, Aug 03 2018

			Triangle begins:
        1
        1       1
        5       3       1
       84      23       6       1
     6348     470      65      10       1
  7743728   39598    1575     145      15       1

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

First column is A048143. Row sums are A006126.
Cf. A000272, A001187, A013922, A030019, A134954, A143543, A275307, A293510.
Cf. A317634, A317635, A317671, A317672, A317677.

blg={1,1,5,84,6348,7743728,2414572893530,56130437190053299918162} (*A048143*);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Product[blg[[Length[s]]],{s,spn}],{spn,Select[sps[Range[n]],Length[#]==k&]}],{n,Length[blg]},{k,n}]

A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 3, 17, 140, 1524, 20673, 336259, 6382302, 138525780, 3384988809, 91976158434, 2751122721402, 89833276321440, 3179852538140115, 121287919647418118, 4959343701136929850, 216406753768138678671, 10037782414506891597734, 493175891246093032826160

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
P. S. Kolesnikov and B. K. Sartayev, On the special identities of Gelfand--Dorfman algebras, arXiv:2105.13815 [math.RA], 2021.
Gus Wiseman, The a(4) = 140 labeled connected graphs with 4 edges.

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A275421, A291842 (planar case), A322114, A322115.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n+1],{2}],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,6}]

Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(8) and beyond from Andrew Howroyd, Nov 28 2018

A322337 Number of strict 2-edge-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 10, 5, 11, 1, 18, 3, 17, 8, 22, 3, 35, 5, 32, 17, 39, 16, 59, 14, 58, 33, 75, 28, 103, 35, 106, 71, 125, 63, 174, 81, 192, 127, 220, 130, 294, 170, 325, 237, 378, 257, 504

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

An integer partition is 2-edge-connected if the hypergraph of prime factorizations of its parts is connected and cannot be disconnected by removing any single part.

			The a(24) = 18 strict 2-edge-connected integer partitions of 24:
  (15,9)   (10,8,6)   (10,8,4,2)
  (16,8)   (12,8,4)   (12,6,4,2)
  (18,6)   (12,9,3)
  (20,4)   (14,6,4)
  (21,3)   (14,8,2)
  (22,2)   (15,6,3)
  (14,10)  (16,6,2)
           (18,4,2)
           (12,10,2)

Wikipedia, k-edge-connected graph

Cf. A007718, A013922, A054921, A095983, A218970, A275307, A286518, A304714, A304716, A305078, A305079, A322335, A322336.

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]]]]]]]]];
twoedQ[sys_]:=And[Length[csm[sys]]==1,And@@Table[Length[csm[Delete[sys,i]]]==1,{i,Length[sys]}]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,twoedQ[primeMS/@#]]&]],{n,30}]

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}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A004115 Number of unlabeled rooted nonseparable graphs with n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A317674 Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A322337 Number of strict 2-edge-connected integer partitions of n.

Views