cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A372174 Number of unlabeled simple graphs covering n vertices with a unique triangle.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 16, 79, 424, 3098, 28616
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2024

Keywords

Comments

The labeled version is A372171.

Links

Crossrefs

The non-covering version is column k = 1 of A263340, labeled A372170.
Case of A370167 with a unique triangle, labeled A054548.
For no triangles we have A372169, labeled A372168 (non-covering A213434).
The labeled version is A372171, column k = 1 of A372167.
Column k = 1 of A372173, labeled A372167.
For cycles (not just triangles) we have A372191, labeled A372195.
The non-covering version is A372194, labeled A372172.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A002494 counts unlabeled covering graphs, labeled A006129.
A372176 counts labeled graphs by directed cycles, covering A372175.
Cf. A000055, A053530, A137917, A137918, A140637, A144958, A370169.

Formula

First differences of A372194.

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

Views

Author

Gus Wiseman, Apr 25 2024

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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.

Programs

  • Mathematica
    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}]
  • PARI
    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

Formula

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

Extensions

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

A372195 Number of labeled simple graphs covering n vertices with a unique undirected cycle of length > 2.

Original entry on oeis.org

0, 0, 0, 1, 15, 232, 3945, 75197, 1604974, 38122542, 1000354710, 28790664534, 902783451933, 30658102047787, 1121532291098765, 43985781899812395, 1841621373756094796, 82002075703514947236, 3869941339069299799884, 192976569550677042208068, 10139553075163838030949495
Offset: 0

Views

Author

Gus Wiseman, Apr 25 2024

Keywords

Comments

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.

Examples

			The a(4) = 15 graphs:
  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

Links

Crossrefs

For no cycles we have A105784 (for triangles A372168, non-covering A213434), unlabeled A144958 (for triangles A372169).
Counting triangles instead of cycles gives A372171 (non-covering A372172), unlabeled A372174 (non-covering A372194).
The unlabeled version is A372191, non-covering A236570.
The non-covering version is A372193, column k = 1 of A372176.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A002807 counts cycles in a complete graph.
A006129 counts labeled graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A372167 counts covering graphs by triangles (non-covering A372170), unlabeled A372173 (non-covering A263340).
Cf. A000169, A000272, A053530, A054548, A137916, A137917, A367863, A368597.

Programs

  • Mathematica
    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}]],Union@@#==Range[n]&&Length[cyc[#]]==2&]],{n,0,5}]
  • PARI
    seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-w-w^2/2-x)*(-log(1+w)/2 + w/2 - w^2/4)), -n-1)} \\ Andrew Howroyd, Jul 31 2024

Formula

Inverse binomial transform of A372193. - Andrew Howroyd, Jul 31 2024

Extensions

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

A372175 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly 2k directed cycles of length > 2.

Original entry on oeis.org

1, 0, 1, 3, 1, 19, 15, 0, 6, 0, 0, 0, 1, 155, 232, 15, 190, 0, 0, 70, 50, 0, 0, 0, 0, 30, 15, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2024

Keywords

Comments

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

Examples

			Triangle begins (zeros shown as dots):
  1
  .
  1
  3 1
  19 15 . 6 ... 1
  155 232 15 190 .. 70 50 .... 30 15 .......... 10 .............. 1
Row n = 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  12,13,24,34     13,14,23,24,34
  12,14,23  12,14,23,24
  12,14,34  12,14,23,34
  12,23,24  12,14,24,34
  12,23,34  12,23,24,34
  12,24,34  13,14,23,24
  13,14,23  13,14,23,34
  13,14,24  13,14,24,34
  13,23,24  13,23,24,34
  13,23,34  14,23,24,34
  13,24,34
  14,23,24
  14,23,34
  14,24,34

Crossrefs

Row lengths are A002807 + 1.
Row sums are A006129, unlabeled A002494.
Column k = 0 is A105784 (for triangles A372168, non-covering A213434), unlabeled A144958 (for triangles A372169).
Counting triangles instead of cycles gives A372167 (non-covering A372170), unlabeled A372173 (non-covering A263340).
The non-covering version is A372176.
Column k = 1 is A372195 (non-covering A372193, for triangles A372171), unlabeled A372191 (non-covering A236570, for triangles A372174).
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000272, A053530, A121251, A137916, A137917, A372172, A372194.

Programs

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

A372194 Number of unlabeled graphs with n vertices and a unique triangle.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 23, 102, 526, 3624, 32240, 382095, 5986945
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2024

Keywords

Comments

The labeled version is A372172.

Examples

			Representatives of the a(3) = 1 through a(6) = 23 graphs:
    12,13,23    12,13,23       12,13,23             12,13,23
                14,23,24,34    12,34,35,45          12,34,35,45
                               14,23,24,34          14,23,24,34
                               12,25,34,35,45       12,25,34,35,45
                               14,25,34,35,45       12,36,45,46,56
                               15,25,34,35,45       13,23,45,46,56
                               12,14,25,34,35,45    14,25,34,35,45
                                                    15,25,34,35,45
                                                    12,14,25,34,35,45
                                                    12,23,36,45,46,56
                                                    13,23,36,45,46,56
                                                    13,25,36,45,46,56
                                                    13,26,36,45,46,56
                                                    14,25,36,45,46,56
                                                    15,26,36,45,46,56
                                                    16,26,36,45,46,56
                                                    12,13,25,36,45,46,56
                                                    12,13,26,36,45,46,56
                                                    13,23,25,36,45,46,56
                                                    14,23,25,36,45,46,56
                                                    16,23,25,36,45,46,56
                                                    13,14,23,25,36,45,46,56
                                                    13,15,23,25,36,45,46,56

Links

Crossrefs

For no triangles we have A006785, covering A372169.
Column k = 1 of A263340, covering A372173.
The labeled version is A372172.
The covering case is A372174, labeled A372171.
For all cycles (not just triangles): A236570, A372193, A372191, A372195.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A002494 counts unlabeled covering graphs, labeled A006129.
A372176 counts labeled graphs by directed cycles, covering A372175.
Cf. A000055, A137917, A137918, A140637, A144958, A213434, A372168, A372170.

Programs

Formula

First differences are A372174.

Extensions

a(11)-a(12) added by Georg Grasegger, Aug 03 2024
Previous Showing 11-15 of 15 results.

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