A372174 -id:A372174 - cp's OEIS frontend

A144958 Number of unlabeled acyclic graphs covering n vertices.

1, 0, 1, 1, 3, 4, 10, 17, 39, 77, 176, 381, 891, 2057, 4941, 11915, 29391, 73058, 184236, 468330, 1202349, 3108760, 8097518, 21218776, 55925742, 148146312, 394300662, 1053929982, 2828250002, 7617271738, 20584886435, 55802753243

Offset: 0

Washington Bomfim, Sep 27 2008

nonn

a(n) is the number of forests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A005195(n-1) counts the forests with one or more isolated nodes.

The labeled version is A105784. The connected case is A000055. This is the covering case of A005195. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of non-isomorphic representatives of the a(0) = 1 through a(5) = 4 forests:
  {}    .    {12}    {13,23}    {12,34}       {12,35,45}
                                {13,24,34}    {13,24,35,45}
                                {14,24,34}    {14,25,35,45}
                                              {15,25,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The connected case is A000055.
This is the covering case of A005195, labeled A001858.
The labeled version is A105784.
For triangles instead of cycles we have A372169, non-covering A006785.
Unique cycle: A372191 (lab A372195), non-covering A236570 (lab A372193).
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000272, A053530, A054548, A137917, A144959, A372174.

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}]]];
cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y,{k}],And@@Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&],{k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Union[Union[brute/@Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[cyc[#]]==0&]]]],{n,0,5}] (* Gus Wiseman, Apr 29 2024 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n), v=EulerT(Vec(t - t^2/2 + subst(t,x,x^2)/2))); concat([1,0], vector(#v-1, i, v[i+1]-v[i]))} \\ Andrew Howroyd, Aug 01 2024

a(n) = A005195(n) - A005195(n-1).

Name changed and 1 prepended by Gus Wiseman, Apr 29 2024.

A263340 Triangle read by rows: T(n,k) is the number of graphs with n vertices containing k triangles.

Original entry on oeis.org

1, 1, 2, 3, 1, 7, 2, 1, 0, 1, 14, 7, 5, 2, 3, 1, 0, 1, 0, 0, 1, 38, 23, 28, 14, 18, 9, 7, 5, 4, 1, 4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 107, 102, 141, 117, 123, 92, 80, 63, 49, 35, 35, 23, 15, 17, 10, 4, 9, 5, 2, 3, 3, 2, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Christian Stump, Oct 15 2015

Row sums give A000088.
First column is A006785.
Row lengths are 1 + binomial(n,3). - Geoffrey Critzer, Apr 13 2017

			Triangle begins:
  1;
  1;
  2;
  3,1;
  7,2,1,0,1;
  14,7,5,2,3,1,0,1,0,0,1;
  38,23,28,14,18,9,7,5,4,1,4,1,1,1,0,0,1,0,0,0,1;
  ...

Pontus von Brömssen, Rows n = 0..10, flattened
FindStat - Combinatorial Statistic Finder, The number of triangles of a graph.
Gus Wiseman, The graphs counted under row n = 5.

Row sums are A000088, labeled A006125.
Column k = 0 is A006785 (lab A213434), covering A372169 (lab A372168).
Counting edges gives A008406 (lab A084546), covering A370167 (lab A054548).
Row lengths are A050407.
The labeled version is A372170, covering A372167.
The covering case is A372173, sums A002494, labeled A006129.
Column k = 1 is A372194 (lab A372172), covering A372174 (lab A372171).
A001858 counts acyclic graphs, unlabeled A005195.
A372176 counts labeled graphs by directed cycles, covering A372175.
Cf. A000055, A053530, A137917, A137918, A140637, A144958, A322700, A370169.

Table[Table[Count[Table[Tr[MatrixPower[AdjacencyMatrix[GraphData[{n, i}]], 3]]/6, {i, 1, NumberOfGraphs[n]}], k], {k, 0, Binomial[n, 3]}], {n, 1, 7}] (* Geoffrey Critzer, Apr 13 2017 *)

Row 7 from Geoffrey Critzer, Apr 13 2017

T(0,0)=1 prepended by Alois P. Heinz, Apr 13 2017

A372169 Number of unlabeled triangle-free graphs covering n vertices.

Original entry on oeis.org

1, 0, 1, 1, 4, 7, 24, 69, 303, 1487, 10275, 92899, 1157109, 19534822, 447074367, 13764681083, 567227701549, 31139379910949

Offset: 0

Gus Wiseman, Apr 23 2024

The labeled version is A372168.

			Non-isomorphic representatives of the a(5) = 7 graphs:
  12-35-45
  13-24-35-45
  14-25-35-45
  15-25-35-45
  12-13-24-35-45
  15-23-24-35-45
  13-14-23-24-35-45

Eric Weisstein's World of Mathematics, Triangle-Free Graph
Gus Wiseman, The a(6) = 24 unlabeled triangle-free graphs covering 6 vertices.

Dominated by A002494, labeled A006129.
Covering case of A006785, labeled A213434.
The connected case is A024607.
For all cycles (not just triangles) we have A144958, labeled A105784.
The labeled version is A372168.
For a unique triangle (labeled) we have A372171, non-covering A372172.
Column k = 0 of A372173, labeled A372167.
For a unique triangle (unlabeled) we have A372174, non-covering A372194.
A001858 counts acyclic graphs, unlabeled A005195.
A006125 counts simple graphs, unlabeled A000088.
A054548 counts covering graphs by number of edges, unlabeled A370167.
A372170 counts graphs by triangles, unlabeled A263340.
Cf. A000055, A000272, A053530, A121251, A322700, A372175, A372191.

First differences of A006785.

A372191 Number of unlabeled simple graphs covering n vertices with a unique undirected cycle of length > 2.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 16, 43, 117, 319, 875, 2409, 6692, 18614, 52099, 146186, 411720, 1162295, 3289994, 9330913, 26517036, 75481622, 215201178, 614398459, 1756392061, 5026955216, 14403488345, 41311616835, 118601561506, 340795908579, 980078195995

Offset: 0

Gus Wiseman, Apr 27 2024

nonn

An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.

Andrew Howroyd, Table of n, a(n) for n = 0..500
Gus Wiseman, The a(3) = 1 through a(6) = 16 covering graphs with a unique cycle.

For no cycles we have A144958 (non-covering A005195), labeled A105784 (non-covering A001858).
Counting triangles instead of cycles gives A372174 (non-covering A372194), labeled A372171 (non-covering A372172).
The non-covering version is A236570, labeled A372193.
The labeled version is A372195, column k = 1 of A372175.
A002807 counts cycles in a complete graph.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A372167 counts graphs by triangles, non-covering A372170.
A372173 counts unlabeled graphs by triangles (non-covering A263340).
A372176 counts labeled graphs by directed cycles.
Cf. A000055, A137916, A137917, A137918, A140637, A370167, A370169.

First differences of A236570.

a(7) onwards from Andrew Howroyd, Jul 31 2024

A372167 Irregular triangle read by rows where T(n,k) is the number of simple graphs covering n vertices with exactly k triangles, 0 <= k <= binomial(n,3).

Original entry on oeis.org

1, 0, 1, 3, 1, 22, 12, 6, 0, 1, 237, 220, 165, 70, 35, 30, 0, 10, 0, 0, 1, 3961, 5460, 5830, 4140, 2805, 2112, 1230, 720, 600, 180, 230, 60, 45, 60, 0, 0, 15, 0, 0, 0, 1, 99900, 191975, 269220, 272055, 240485, 207095, 166005, 121530, 98770, 65905, 48503, 37065, 20055, 17570, 11445, 6552, 4410, 3570, 1680, 1785, 147, 735, 455, 140, 0, 105, 105, 0, 0, 0, 21, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 23 2024

			Triangle begins:
    1
    0
    1
    3    1
   22   12    6    0    1
  237  220  165   70   35   30    0   10    0    0    1
  ...
Row k = 4 counts the following graphs:
  12-34      12-13-14-23  12-13-14-23-24  .  12-13-14-23-24-34
  13-24      12-13-14-24  12-13-14-23-34
  14-23      12-13-14-34  12-13-14-24-34
  12-13-14   12-13-23-24  12-13-23-24-34
  12-13-24   12-13-23-34  12-14-23-24-34
  12-13-34   12-14-23-24  13-14-23-24-34
  12-14-23   12-14-24-34
  12-14-34   12-23-24-34
  12-23-24   13-14-23-34
  12-23-34   13-14-24-34
  12-24-34   13-23-24-34
  13-14-23   14-23-24-34
  13-14-24
  13-23-24
  13-23-34
  13-24-34
  14-23-24
  14-23-34
  14-24-34
  12-13-24-34
  12-14-23-34
  13-14-23-24

Andrew Howroyd, Table of n, a(n) for n = 0..340 (rows 0..10)
Gus Wiseman, All simple graphs covering {1,2,3,4} grouped by number of triangles.

Row sums are A006129, unlabeled A002494.
Row lengths are A050407.
Counting edges instead of triangles gives A054548, unlabeled A370167.
Column k = 0 is A372168 (non-covering A213434), unlabeled A372169.
Covering case of A372170, unlabeled A263340.
Column k = 1 is A372171 (non-covering A372172), unlabeled A372174.
The unlabeled version is A372173.
For all cycles (not just triangles) we have A372175, non-covering A372176.
A001858 counts acyclic graphs, unlabeled A005195.
A006125 counts simple graphs, unlabeled A000088.
A105784 counts acyclic covering graphs, unlabeled A144958.
Cf. A000272, A053530, A121251, A137916, A367868, A369199, A372191.

cys[y_]:=Select[Subsets[Union@@y,{3}], MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Length[cys[#]]==k&]], {n,0,5},{k,0,Binomial[n,3]}]

Inverse binomial transform of columns of A372170.

a(42) onwards from Andrew Howroyd, Dec 29 2024

A372171 Number of labeled simple graphs covering n vertices with a unique triangle.

Original entry on oeis.org

0, 0, 0, 1, 12, 220, 5460, 191975, 9596160, 683389812, 69270116040

Offset: 0

Gus Wiseman, Apr 24 2024

The unlabeled version is A372174.

			The a(4) = 12 graphs:
  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

Column k = 1 of A372167, unlabeled A372173.
For no triangles we have A372168 (non-covering A213434), unlabeled A372169.
The non-covering case is A372172, unlabeled A372194.
The unlabeled version is A372174.
For all cycles (not just triangles) we have A372195, non-covering A372193.
A001858 counts acyclic graphs, unlabeled A005195.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494
A054548 counts labeled covering graphs by edges, unlabeled A370167.
A105784 counts acyclic covering graphs, unlabeled A144958.
A372170 counts graphs by triangles, unlabeled A263340.
A372175 counts covering graphs by cycles, non-covering A372176.
Cf. A000272, A053530, A121251, A137916, A367868, A369199, A372191.

cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Union@@#==Range[n]&&Length[cys[#]]==1&]],{n,0,5}]

Inverse binomial transform of A372172.

a(7)-a(10) from Andrew Howroyd, Aug 01 2024

A372173 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k triangles, 0 <= k <= binomial(n,3).

Original entry on oeis.org

1, 0, 1, 1, 1, 4, 1, 1, 0, 1, 7, 5, 4, 2, 2, 1, 0, 1, 0, 0, 1, 24, 16, 23, 12, 15, 8, 7, 4, 4, 1, 3, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 69, 79, 113, 103, 105, 83, 73, 58, 45, 34, 31, 22, 14, 16, 10, 4, 8, 5, 2, 3, 2, 2, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 23 2024

			Triangle begins:
  1
  0
  1
  1 1
  4 1 1 0 1
  7 5 4 2 2 1 0 1 0 0 1

Andrew Howroyd, Table of n, a(n) for n = 0..340 (rows 0..10)
Gus Wiseman, All unlabeled simple graphs covering 5 vertices, grouped by number of triangles.

Row sums are A002494, labeled A006129.
Row lengths are A050407.
The non-covering version is A263340, labeled A372170.
Counting edges instead of triangles gives A370167, labeled A054548.
The labeled version is A372167.
Column k = 0 is A372169, labeled A372168 (non-covering A213434).
Column k = 1 is A372174, labeled A372171.
Column k = 1 is also the covering case of A372194, labeled A372172.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A372176 counts labeled graphs by directed cycles, covering A372175.
Cf. A000055, A053530, A137917, A137918, A140637, A144958, A322700, A370169.

a(21) onwards from Andrew Howroyd, Dec 29 2024

A372168 Number of triangle-free simple labeled graphs covering n vertices.

Original entry on oeis.org

1, 0, 1, 3, 22, 237, 3961, 99900, 3757153, 208571691, 16945953790, 1999844518737, 340422874696873, 83041703920313712, 28850117307732482737, 14191512425207950473867, 9829313296102303971441502

Offset: 0

Gus Wiseman, Apr 23 2024

The unlabeled version is A372169.

			The a(4) = 22 graphs are:
  12-34
  13-24
  14-23
  12-13-14
  12-13-24
  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
  12-13-24-34
  12-14-23-34
  13-14-23-24

Eric Weisstein's World of Mathematics, Triangle-Free Graph

Dominated by A006129, unlabeled A002494.
For all cycles (not just triangles) we have A105784, unlabeled A144958.
Covering case of A213434 (column k = 0 of A372170, unlabeled A263340).
The connected case is A345218, unlabeled A024607.
Column k = 0 of A372167, unlabeled A372173.
The unlabeled version is A372169.
For a unique triangle we have A372171, non-covering A372172.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A054548 counts covering graphs by number of edges, unlabeled A370167.
Cf. A000272, A053530, A121251, A367862, A367863, A372191, A372174, A372175.

cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Union@@#==Range[n]&&Length[cys[#]]==0&]],{n,0,5}]

Binomial transform is A213434.

A372172 Number of labeled simple graphs on n vertices with exactly one triangle.

Original entry on oeis.org

0, 0, 0, 1, 16, 290, 6980, 235270, 11298056, 777154308, 76560083040

Offset: 0

Gus Wiseman, Apr 24 2024

The unlabeled version is A372194.

			The a(4) = 16 graphs:
  12,13,23
  12,14,24
  13,14,34
  23,24,34
  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

For no triangles we have A213434, covering A372168 (unlabeled A372169).
Column k = 1 of A372170, unlabeled A263340.
The covering case is A372171, unlabeled A372174.
For all cycles (not just triangles) we have A372193, covering A372195.
The unlabeled version is A372194.
A001858 counts acyclic graphs, unlabeled A005195.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494
A054548 counts labeled covering graphs by edges, unlabeled A370167.
A372167 counts covering graphs by triangles, unlabeled A372173.
Cf. A000272, A105784, A121251, A137916, A236570, A372176.

cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cys[#]]==1&]],{n,0,5}]

Binomial transform of A372171.

a(8)-a(10) from Andrew Howroyd, Aug 01 2024

A372193 Number of labeled simple graphs on n vertices with a unique cycle of length > 2.

Original entry on oeis.org

0, 0, 0, 1, 19, 317, 5582, 108244, 2331108, 55636986, 1463717784, 42182876763, 1323539651164, 44955519539963, 1644461582317560, 64481138409909506, 2698923588248208224, 120133276796015812548, 5667351458582453925696, 282496750694780020437765, 14837506263979393796687088

Offset: 0

Gus Wiseman, Apr 25 2024

nonn

An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.

			The a(4) = 19 graphs:
  12,13,23
  12,14,24
  13,14,34
  23,24,34
  12,13,14,23
  12,13,14,24
  12,13,14,34
  12,13,23,24
  12,13,23,34
  12,13,24,34
  12,14,23,24
  12,14,23,34
  12,14,24,34
  12,23,24,34
  13,14,23,24
  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..200

For no cycles we have A001858 (covering A105784), unlabeled A005195 (covering A144958).
Counting triangles instead of cycles gives A372172 (non-covering A372171), unlabeled A372194 (non-covering A372174).
The unlabeled version is A236570, non-covering A372191.
The covering case is A372195, column k = 1 of A372175.
A000088 counts unlabeled graphs, labeled A006125.
A002807 counts cycles in a complete graph.
A006129 counts labeled graphs, unlabeled A002494.
A372167 counts graphs by triangles, non-covering A372170.
A372173 counts unlabeled graphs by triangles, non-covering A263340.
Cf. A000272, A054548, A057500, A121251, A137916, A213434, A322661, A372169, A372176.

cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations /@ Subsets[Union@@y,{k}],And @@ Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&], {k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cyc[#]]==2&]],{n,0,5}]

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-w-w^2/2)*(-log(1+w)/2 + w/2 - w^2/4)), -n-1)} \\ Andrew Howroyd, Jul 31 2024

E.g.f.: B(x)*C(x) where B(x) is the e.g.f. of A057500 and C(x) is the e.g.f. of A001858. - Andrew Howroyd, Jul 31 2024

a(7) onwards from Andrew Howroyd, Jul 31 2024

cp's OEIS Frontend

A144958 Number of unlabeled acyclic graphs covering n vertices.

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A263340 Triangle read by rows: T(n,k) is the number of graphs with n vertices containing k triangles.

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A372169 Number of unlabeled triangle-free graphs covering n vertices.

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A372191 Number of unlabeled simple graphs covering n vertices with a unique undirected cycle of length > 2.

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A372167 Irregular triangle read by rows where T(n,k) is the number of simple graphs covering n vertices with exactly k triangles, 0 <= k <= binomial(n,3).

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A372171 Number of labeled simple graphs covering n vertices with a unique triangle.

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A372173 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k triangles, 0 <= k <= binomial(n,3).

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A372168 Number of triangle-free simple labeled graphs covering n vertices.

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