A000664 -id:A000664 - cp's OEIS frontend

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

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

A053418 Number of unlabeled directed graphs with n arcs and no isolated vertices.

Original entry on oeis.org

1, 1, 5, 17, 80, 365, 1981, 11222, 69511, 455663, 3169244, 23170347, 177513359, 1418920570, 11798710013, 101778754655, 908722427531, 8380602471646, 79692654473866, 780142956502644, 7851084073063731, 81120767066417308

Offset: 0

Vladeta Jovovic, Jan 10 2000

nonn

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

The labeled version is A121252.
Column sums of A350908.
Cf. A000273, A000664, A053454, A053598 (by # of nodes).

Euler transform of A053454. - Andrew Howroyd, Jan 28 2022

Edited and extended by Max Alekseyev, Sep 18 2009

A241706 Number of simple connected graphs on n nodes with diameter 2.

Original entry on oeis.org

0, 0, 1, 4, 14, 59, 373, 4154, 91518, 4116896

Offset: 1

Travis Hoppe and Anna Petrone, Apr 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 [But beware errors] (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Graph Diameter

Column k=2 of A294522.

Simple connected graphs of diameter k: A241706, A241707, A241708, A241709, A241710.

A201922 Triangle read by rows: T(n,m) = number of unlabeled graphs on n nodes with m connected components, m = 1,2,...,n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 21, 8, 3, 1, 1, 112, 30, 9, 3, 1, 1, 853, 145, 32, 9, 3, 1, 1, 11117, 1028, 154, 33, 9, 3, 1, 1, 261080, 12320, 1065, 156, 33, 9, 3, 1, 1, 11716571, 274806, 12513, 1074, 157, 33, 9, 3, 1, 1, 1006700565, 12007355, 276114, 12550, 1076, 157, 33, 9, 3, 1, 1

Offset: 1

Max Alekseyev, Dec 06 2011

			Triangle starts:
    1
    1   1
    2   1   1
    6   3   1   1
   21   8   3   1   1
  112  30   9   3   1   1
  853 145  32   9   3   1   1 ...

Alois P. Heinz, Rows n = 1..75, flattened
P. Flajolet, R. Sedgewick, Analytic combinatorics, Theorem I.1 (Multiset)
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 82
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 5 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

Cf. A001349 (first column), A000088 (row sum), A201968 (limits in the diagonals), A106240, A274934 (2nd column).

nn=10; c=(A000088=Table[NumberOfGraphs[n], {n,0,nn}]; f[x_] = 1-Product[1/(1-x^k)^a[k], {k,1,nn}]; a[0]=a[1]=a[2]=1; coes=CoefficientList[Series[f[x], {x,0,nn}], x]; sol=First[Solve[Thread[Rest[coes+A000088]==0]]]; Table[a[n], {n,0,nn}]/.sol); f[list_]:=Select[list,#>0&]; g=Product[1/(1-y x^n)^c[[n+1]], {n,1,nn}]; Map[f, Drop[CoefficientList[Series[g, {x,0,nn}], {x,y}],1]] //Flatten (* Geoffrey Critzer, Apr 19 2012  (c in above Mma code is given by Jean Francois Alcover in A001349) *)

T(n,m) = sum over the partitions of n with m parts: 1*K1 + 2*K2 + ... + n*Kn = n, K1 + K2 + ... + Kn = m, of Product_{i=1..n} binomial(A001349(i) + Ki - 1, Ki).

O.g.f.: Product_{n>=1} 1/(1 - y*x^n)^A001349(n). - Geoffrey Critzer, Apr 19 2012

A241707 Number of simple connected graphs on n nodes with diameter 3.

Original entry on oeis.org

0, 0, 0, 1, 5, 43, 387, 5797, 148229, 6959721

Offset: 1

Travis Hoppe and Anna Petrone, Apr 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 [But beware errors] (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Graph Diameter

Column k=3 of A294522.

Simple connected graphs of diameter k: A241706, A241707, A241708, A241709, A241710.

A263859 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of posets with n elements and rank k (or depth k+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 20, 31, 10, 1, 1, 55, 162, 84, 15, 1, 1, 163, 940, 734, 185, 21, 1, 1, 556, 6372, 7305, 2380, 356, 28, 1, 1, 2222, 52336, 86683, 35070, 6259, 623, 36, 1, 1, 10765, 534741, 1261371, 619489, 125597, 14258, 1016, 45, 1

Offset: 1

Christian Stump, Oct 28 2015

Row sums give A000112, n >= 1.

The rank of a poset is the number of cover relations in a maximal chain.

			Triangle begins:
1,
1,1,
1,3,1,
1,8,6,1,
1,20,31,10,1,
1,55,162,84,15,1,
1,163,940,734,185,21,1,
1,556,6372,7305,2380,356,28,1,
1,2222,52336,86683,35070,6259,623,36,1,
1,10765,534741,1261371,619489,125597,14258,1016,45,1,
...

R. J. Mathar, Table of n, a(n) for n = 1..136
G. Brinkmann and B. D. McKay, Posets on up to 16 Points [On Brendan McKay's home page]
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179.
FindStat - Combinatorial Statistic Finder, The rank of the poset.
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

Cf. A000112 (row sums), A342500 (connected).

More terms from Brinkmann-McKay (2002) added by N. J. A. Sloane, Mar 18 2017

A303832 The number of edge-rooted unlabeled connected graphs with n edges.

Original entry on oeis.org

1, 1, 4, 10, 32, 101, 346, 1220, 4517, 17338, 69107, 285009, 1215015, 5344224, 24223641, 113001129, 541913075, 2668817544, 13484234188, 69831773559, 370361639587, 2009988998148, 11153858854425, 63242354288220, 366140089188603, 2163036956456422, 13031489297543608

Offset: 1

R. J. Mathar, May 04 2018

nonn

			a(1)=1: the connected graph with 1 edge (which is rooted).
a(2)=1: the connected graph with 2 edges (one rooted).
a(3)=4: the triangle graph with one choice of rooting, the linear tree with either the middle or a terminating edge rooted, the star graph with one edge rooted.

Cf. A126133 (not necessarily connected), A000664, A303830 (by number of nodes).

Cf. A002905, A339039, A339040, A339041, A339044.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/(2*G(2*n, x+A, [])*(1+x)))} \\ Andrew Howroyd, Nov 21 2020

G.f. A(x) satisfies: A(x)*A000664(x) = A126133(x).

A322147 Regular triangle read by rows where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 1, 1, 0, 2, 3, 0, 1, 10, 16, 0, 0, 12, 79, 125, 0, 0, 6, 162, 847, 1296, 0, 0, 1, 179, 2565, 11436, 16807, 0, 0, 0, 116, 4615, 47100, 185944, 262144, 0, 0, 0, 45, 5540, 121185, 987567, 3533720, 4782969, 0, 0, 0, 10, 4720, 220075, 3376450, 23315936, 76826061, 100000000

Offset: 0

Gus Wiseman, Nov 28 2018

			Triangle begins:
  1
  1     1
  0     2     3
  0     1    10    16
  0     0    12    79   125
  0     0     6   162   847  1296
  0     0     1   179  2565 11436 16807

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A322151. Last column is A000272.
Column sums are A062740.
Cf. A000664, A007718, A007719, A054923, A191646, A275421, A321254, A322114, A322115, A322137.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
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[If[n==0,1,Length[Select[Subsets[multsubs[Range[k],2],{n}],And[Union@@#==Range[k],Length[csm[#]]==1]&]]],{n,0,6},{k,1,n+1}]

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}
M(n)={Mat([Col(p, -(n+1)) | p<-Connected(vector(2*n, j, (1 + x + O(x*x^n) )^binomial(j+1,2)))[1..n+1]])}
{ my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ Andrew Howroyd, Nov 29 2018

Terms a(28) and beyond from Andrew Howroyd, Nov 29 2018

A322151 Number of labeled connected graphs with loops with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 2, 5, 27, 216, 2311, 30988, 499919, 9431026, 203743252, 4960335470, 134382267082, 4009794148101, 130668970606412, 4617468180528235, 175867725701333896, 7182126650899080024, 313063334893103361130, 14507460736615554141354, 712192629608088061633746

Offset: 0

Gus Wiseman, Nov 28 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A322147. The unlabeled version is A191970.

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A291842 (planar case), A321254, A322114, A322115.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
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[multsubs[Range[n+1],2],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,5}]

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+1,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 28 2018

A339040 Number of unlabeled connected simple graphs with n edges rooted at two noninterchangeable vertices.

Original entry on oeis.org

1, 3, 10, 35, 125, 460, 1747, 6830, 27502, 113987, 485971, 2129956, 9591009, 44341610, 210345962, 1023182861, 5100235807, 26035673051, 136023990102, 726877123975, 3970461069738, 22156281667277, 126234185382902, 733899631974167, 4351500789211840

Offset: 1

Andrew Howroyd, Nov 20 2020

nonn

Cf. A000664, A002905, A303832, A304074, A339039, A339041, A339042, A339044, A339063.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A, [])); Vec(G(2*n, x+A, [1, 1])/g - (G(2*n, x+A, [1])/g)^2)}

G.f.: f(x)/g(x) - r(x)^2 where f(x), g(x) and r(x) are the g.f.'s of A339063, A000664 and A339039.

cp's OEIS Frontend

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

A053418 Number of unlabeled directed graphs with n arcs and no isolated vertices.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A241706 Number of simple connected graphs on n nodes with diameter 2.

Views

Author

Keywords

Links

Crossrefs

A201922 Triangle read by rows: T(n,m) = number of unlabeled graphs on n nodes with m connected components, m = 1,2,...,n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A241707 Number of simple connected graphs on n nodes with diameter 3.

Views

Author

Keywords

Links

Crossrefs

A263859 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of posets with n elements and rank k (or depth k+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A303832 The number of edge-rooted unlabeled connected graphs with n edges.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A322147 Regular triangle read by rows where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A322151 Number of labeled connected graphs with loops with n edges (the vertices are {1,2,...,k} for some k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions