A372170 -id:A372170 - cp's OEIS frontend

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

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.

A372176 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices with exactly 2k directed cycles of length > 2.

Original entry on oeis.org

1, 1, 2, 7, 1, 38, 19, 0, 6, 0, 0, 0, 1, 291, 317, 15, 220, 0, 0, 70, 55, 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

Gus Wiseman, Apr 25 2024

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.

			Triangle begins (zeros shown as dots):
   1
   1
   2
   7 1
   38 19 . 6 ... 1
   291 317 15 220 .. 70 55 .... 30 15 ........ 10 ............... 1
The T(4,3) = 6 graphs:
  12,13,14,23,24
  12,13,14,23,34
  12,13,14,24,34
  12,13,23,24,34
  12,14,23,24,34
  13,14,23,24,34

Column k = 0 is A001858 (unlabeled A005195), covering A105784.
Row lengths are A002807 + 1.
Row sums are A006125, unlabeled A000088.
Counting edges instead of cycles gives A084546 (covering A054548), unlabeled A008406 (covering A370167).
Counting triangles instead of cycles gives A372170 (covering A372167), unlabeled A263340 (covering A372173).
The covering case is A372175.
Column k = 1 is A372193 (covering A372195), unlabeled A236570.
A006129 counts graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000272, A053530, A137916, A144958, A213434, A367863, A372168, A372169, A372171, A372172, A372191.

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[#]]==2k&]], {n,0,4}, {k,0,Length[cyc[Subsets[Range[n],{2}]]]/2}]

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

A105784 Number of different forests of unrooted trees, without isolated vertices, on n labeled nodes.

Original entry on oeis.org

0, 1, 3, 19, 155, 1641, 21427, 334377, 6085683, 126745435, 2975448641, 77779634571, 2241339267037, 70604384569005, 2414086713172695, 89049201691604881, 3525160713653081279, 149075374211881719939, 6707440248292609651513, 319946143503599791200675

Offset: 1

Washington Bomfim, Apr 21 2005

nonn

Number of labeled acyclic graphs covering n vertices. The unlabeled version is A144958. This is the covering case A001858. The connected case is A000272. - Gus Wiseman, Apr 28 2024

			a(4) = 19 because there are 19 different such forests on 4 labeled nodes: 4^2 are trees, 3 have two trees and none can have more than two trees.
From _Gus Wiseman_, Apr 28 2024: (Start)
Edge-sets of the a(2) = 1 through a(4) = 19 forests:
    12    12,13    12,34
          12,23    13,24
          13,23    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
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..150

Cf. A033185, A105599.
The connected case is A000272, rooted A000169.
This is the covering case of A001858, unlabeled A005195.
The unlabeled version is A144958.
For triangles instead of cycles we have A372168, covering case of A213434.
For a unique cycle we have A372195, covering case of A372193.
A002807 counts cycles in a complete graph.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A372170 counts simple graphs by triangles, covering A372167.
Cf. A053530, A054548, A137916, A137917, A322661, A367863, A372169, A372171, A372176, A372191.

b:= n-> add(add(binomial(m, j) *binomial(n-1, n-m-j)
        *n^(n-m-j) *(m+j)!/ (-2)^j, j=0..m)/m!, m=0..n):
a:= n-> add(b(k) *(-1)^(n-k) *binomial(n, k), k=0..n):
seq(a(n), n=1..17);  # Alois P. Heinz, Sep 10 2008

Unprotect[Power]; 0^0 = 1; b[n_] := Sum[Sum[Binomial[m, j]*Binomial[n-1, n -m-j]*n^(n-m-j)*(m+j)!/(-2)^j, {j, 0, m}]/m!, {m, 0, n}]; a[n_] := Sum[ b[k]*(-1)^(n-k)*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

a(n)= sum N/D over all the partitions of n: 1K1 + 2K2 + ... + nKn, with smallest part greater than 1, where N = n!*Product_{i=1..n}i^((i-2)Ki) and D = Product_{i=1..n}(Ki!(i!)^Ki).
Inverse binomial transform of A001858. E.g.f.: exp(-x-LambertW(-x) -LambertW(-x)^2/2). - Vladeta Jovovic, Apr 22 2005
a(n) ~ exp(-exp(-1)+1/2) * n^(n-2). - Vaclav Kotesovec, Aug 16 2013

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

cp's OEIS Frontend

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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A372176 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices with exactly 2k directed cycles of length > 2.

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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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A105784 Number of different forests of unrooted trees, without isolated vertices, on n labeled nodes.

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