A054548 -id:A054548 - cp's OEIS frontend

A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

1, 1, 4, 23, 199, 2313, 34015, 606407, 12712643, 306407645, 8346154699, 253476928293, 8490863621050, 310937199521774, 12356288017546937, 529516578044589407, 24339848939829286381, 1194495870124420574751, 62332449791125883072149, 3446265450868329833016605

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A369199.

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

The minimal case is A001862, without loops A053530.
This is the covering case of A066383 and A369196, cf. A369192 and A369193.
The case of equality is A368597, without loops A367863.
The version without loops is A369191.
The connected case is A369197, without loops A129271.
The unlabeled version is A370169, equality A368599, non-covering A368598.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable graphs, covering A367868.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A003465, A005703, A006649, A057500, A062740, A116508, A367862, A367916.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[Union@@#]==n&&Length[#]<=n&]],{n,0,5}]

Inverse binomial transform of A369196.

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

A369142 Number of labeled loop-graphs covering {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 1, 22, 616, 26084, 1885323, 253923163, 66619551326, 34575180977552, 35680008747431929, 73392583275070667841, 301348381377662031986734, 2471956814761854578316988092, 40530184362443276558060719358471, 1328619783326799871747200601484790193

Offset: 0

Gus Wiseman, Jan 20 2024

nonn

Also labeled loop-graphs covering n vertices with at least one connected component containing more edges than vertices.

			The a(0) = 0 through a(3) = 22 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{3},{1,2}}
                         {{1},{2},{3},{1,3}}
                         {{1},{2},{3},{2,3}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,2},{2,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3}}
                         {{1},{3},{1,2},{2,3}}
                         {{1},{3},{1,3},{2,3}}
                         {{2},{3},{1,2},{1,3}}
                         {{2},{3},{1,2},{2,3}}
                         {{2},{3},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{2},{1,2},{1,3},{2,3}}
                         {{3},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{3},{1,2},{2,3}}
                         {{1},{2},{3},{1,3},{2,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3},{2,3}}
                         {{2},{3},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3},{2,3}}

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

The version for a unique choice is A000272, unlabeled A000055.
Without the choice condition we have A006125, unlabeled A000088.
The case without loops is A367868, covering case of A367867.
For exactly n edges we have A368730, covering case of A368596.
The complement is counted by A369140, covering case of A368927.
This is the covering case of A369141.
For n edges and no loops we have A369144, covering A369143.
The unlabeled version is A369147, covering case of A369146.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable graphs, unlabeled A005703.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, connected A062740, unlabeled A322700.
A367902 counts choosable set-systems, complement A367903.
Cf. A000666, A003465, A006649, A116508, A137916, A333331, A368600, A368924, A368984, A369145, A369194.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Union@@#==Range[n]&&Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

Inverse binomial transform of A369141.

a(n) = A322661(n) - A369140(n). - Andrew Howroyd, Feb 02 2024

a(6) onwards from Andrew Howroyd, Feb 02 2024

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.

A372170 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k triangles, 0 <= k <= binomial(n,3).

Original entry on oeis.org

1, 1, 2, 7, 1, 41, 16, 6, 0, 1, 388, 290, 195, 70, 40, 30, 0, 10, 0, 0, 1, 5789, 6980, 6910, 4560, 3030, 2292, 1230, 780, 600, 180, 236, 60, 45, 60, 0, 0, 15, 0, 0, 0, 1, 133501, 235270, 313705, 302505, 260890, 222509, 174615, 126780, 102970, 67165, 50134, 37485, 20370, 17990, 11445, 6552, 4515, 3570, 1680, 1785, 154, 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
     1
     2
     7    1
    41   16    6    0    1
   388  290  195   70   40   30    0   10    0    0    1
   ...
For example, the T(4,1) = 16 graphs are:
  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

Andrew Howroyd, Table of n, a(n) for n = 0..340 (rows 0..10)

Row sums are A006125, covering A006129.
Row lengths are A050407.
Counting edges instead of triangles gives A084546, covering A054548.
Column k = 0 is A213434, covering A372168.
The unlabeled version is A263340.
The covering case is A372167, unlabeled A372173.
Column k = 1 is A372172, covering A372171.
For all cycles (not just triangles) we have A372176, covering A372175.
A001858 counts acyclic graphs, unlabeled A005195.
A367867 counts non-choosable graphs, covering A367868.
A372193 counts unicyclic graphs, unlabeled A236570, covering A372191.
Cf. A000088, A000272, A002494, A006785, A053530, A062734, A121251, A372194.

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[#]]==k&]],{n,0,5},{k,0,Binomial[n,3]}]

Binomial transform of columns of A372167.

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

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}]

A121251 Number of labeled graphs without isolated vertices and with n edges.

Original entry on oeis.org

1, 1, 6, 62, 900, 16824, 384668, 10398480, 324420840, 11472953760, 453518054216, 19815916826160, 948348447031440, 49334804947402800, 2771902062752597520, 167281797371598801136, 10791777047497882651296, 741135302021991803931360, 53983717302568691555767360

Offset: 0

Vladeta Jovovic, Aug 22 2006, Sep 19 2006

a(n) ~ C0*(C1*n)^n, where C0 = 1/2^(1+log(2)/4)/log(2) = 0.63970540489176946794... and C1 = 2/(log(2))^2/exp(1) = 1.5313857152078346894..., [Bender et al.]

E. A. Bender, E. R. Canfield and B. D. McKay, The asymptotic number of labeled

E. A. Bender and E. R. Canfield and B. D. McKay, The asymptotic number of labeled graphs with n vertices, q edges and no isolated vertices, preprint, 1996.
E. A. Bender, E. R. Canfield and B. D. McKay, The asymptotic number of labeled graphs with n vertices, q edges and no isolated vertices, J Combinatorial Theory, Series A, 80 (1997) 124-150.
Kassahun H. Betre, Yan X. Zhang, and Carter Edmond, Pure Simplicial and Clique Complexes with a Fixed Number of Facets, arXiv:2411.12945 [math.CO], 2024. See p. 23.
A. N. Bhavale, B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.

Cf. A006129.

a(n) = Sum_{m>=0} binomial(binomial(m,2),n)/2^(m+1). Column sums of A054548.

More terms from Max Alekseyev, Aug 23 2006

A369140 Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).

Original entry on oeis.org

1, 1, 4, 23, 193, 2133, 29410, 486602, 9395315, 207341153, 5147194204, 141939786588, 4304047703755, 142317774817901, 5095781837539766, 196403997108015332, 8106948166404074281, 356781439557643998591, 16675999433772328981216, 824952192369049982670686

Offset: 0

Gus Wiseman, Jan 20 2024

nonn

These are covering loop-graphs where every connected component has a number of edges less than or equal to the number of vertices in that component. Also covering loop-graphs with at most one cycle (unicyclic) in each connected component.

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

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

For a unique choice we have A000272, covering case of A088957.
Without the choice condition we have A322661, unlabeled A322700.
For exactly n edges we have A333331 (maybe), complement A368596.
The case without loops is A367869, covering case of A133686.
This is the covering case of A368927.
The complement is counted by A369142, covering case of A369141.
The unlabeled version is the first differences of A369145.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A367862 counts graphs with n vertices and n edges, covering A367863.
Cf. A000169, A000666, A003465, A006649, A062740, A116508, A137916, A368924, A368984, A369194.

Table[Length[Select[Subsets[Subsets[Range[n], {1,2}]],Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]],{n,0,5}]

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(exp(-x + 3*t/2 - 3*t^2/4)/sqrt(1-t) ))} \\ Andrew Howroyd, Feb 02 2024

Inverse binomial transform of A368927.
Exponential transform of A369197.
E.g.f.: exp(-x)*exp(3*T(x)/2 - 3*T(x)^2/4)/sqrt(1-T(x)), where T(x) is the e.g.f. of A000169. - Andrew Howroyd, Feb 02 2024

a(6) onwards from Andrew Howroyd, Feb 02 2024

A369191 Number of labeled simple graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 0, 1, 4, 34, 387, 5686, 102084, 2162168, 52693975, 1450876804, 44509105965, 1504709144203, 55563209785167, 2224667253972242, 95984473918245388, 4439157388017620554, 219067678811211857307, 11489425098298623161164, 638159082104453330569185

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A054548.

			The a(0) = 1 through a(3) = 4 graphs:
  {}  .  {{1,2}}  {{1,2},{1,3}}
                  {{1,2},{2,3}}
                  {{1,3},{2,3}}
                  {{1,2},{1,3},{2,3}}

The minimal case is A053530.
The connected case is A129271, unlabeled version A005703.
The case of equality is A367863, covering case of A367862.
This is the covering case of A369192, or A369193 for covered vertices.
The version for loop-graphs is A369194.
The unlabeled version is A370316.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A057500 counts connected graphs with n vertices and n edges.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001429, A003465, A006649, A143543, A116508, A140638, A322661.

Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[Union@@#]==n&&Length[#]<=n&]],{n,0,5}]

Inverse binomial transform of A369193.

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

cp's OEIS Frontend

A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

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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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A369142 Number of labeled loop-graphs covering {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

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

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A372170 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k triangles, 0 <= k <= binomial(n,3).

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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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A121251 Number of labeled graphs without isolated vertices and with n edges.

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A369140 Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).

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