A002807 -id:A002807 - cp's OEIS frontend

A005195 Number of forests with n unlabeled nodes.

1, 1, 2, 3, 6, 10, 20, 37, 76, 153, 329, 710, 1601, 3658, 8599, 20514, 49905, 122963, 307199, 775529, 1977878, 5086638, 13184156, 34402932, 90328674, 238474986, 632775648, 1686705630, 4514955632, 12132227370, 32717113805, 88519867048, 240235675303

Offset: 0

N. J. A. Sloane

Same as "Number of forests with n nodes that are perfect graphs" [see Hougardy]. - N. J. A. Sloane, Dec 04 2015

Number of unlabeled acyclic graphs on n vertices. The labeled version is A001858. The covering case is A144958, connected A000055. - 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) = 10 forests:
  {}  {}  {}    {}       {}          {}
          {12}  {12}     {12}        {12}
                {13,23}  {12,34}     {12,34}
                         {13,23}     {13,23}
                         {13,24,34}  {12,35,45}
                         {14,24,34}  {13,24,34}
                                     {14,24,34}
                                     {13,24,35,45}
                                     {14,25,35,45}
                                     {15,25,35,45}
(End)

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 58-59.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
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.)
Eric Weisstein's World of Mathematics, Forest.
Index entries for sequences related to forests

Cf. A095133 (by number of trees), A136605 (by number of edges).
A diagonal of A144215.
The connected case is A000055.
The labeled version is A001858.
The covering case is A144958, labeled A105784.
For triangles instead of cycles we have A006785, covering A372169.
Unique cycle: A236570 (labeled A372193), covering A372191 (labeled A372195).
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000272, A002807, A054548, A137917, A367863, A372176.

EulerTransform[ seq_List ] := With[{m = Length[seq]}, CoefficientList[ Series[ Times @@ (1/(1 - x^Range[m])^seq), {x, 0, m}], x]];
b[n_] := b[n] = If[n <= 1, n, Sum[ Sum[ d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}]/(n - 1)];
a55[n_] := a55[n] = If[n == 0, 1, b[n] - (Sum[ b[k]*b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2]; A000055 = Table[ a55[n], {n, 1, 31}]; EulerTransform[ A000055 ] (* Jean-François Alcover, Mar 15 2012 *)

Euler transform of A000055: Product_{n>0} (1-x^n)^(-A000055(n)). a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*A000055(d). - Vladeta Jovovic, Sep 05 2002
G.f.: exp(sum_{k>0} B(x^k)/k ), where B(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + ... = C(x)-1 and C is the g.f. for A000055.
a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148..., c = 1.023158422... . - Vaclav Kotesovec, Nov 16 2014
First differences are A144958. - Gus Wiseman, Apr 29 2024

More terms from Vladeta Jovovic, Sep 05 2002

A001858 Number of forests of trees on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 7, 38, 291, 2932, 36961, 561948, 10026505, 205608536, 4767440679, 123373203208, 3525630110107, 110284283006640, 3748357699560961, 137557910094840848, 5421179050350334929, 228359487335194570528, 10239206473040881277575, 486909744862576654283616

Offset: 0

N. J. A. Sloane and Simon Plouffe

The number of integer lattice points in the permutation polytope of {1,2,...,n}. - Max Alekseyev, Jan 26 2010
Equals the number of score sequences for a tournament on n vertices. See Prop. 7 of the article by Bartels et al., or Example 3.1 in the article by Stanley. - David Radcliffe, Aug 02 2022
Number of labeled acyclic graphs on n vertices. The unlabeled version is A005195. The covering case is A105784, connected A000272. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of the a(4) = 38 forests:
  {}  {12}  {12,13}  {12,13,14}
      {13}  {12,14}  {12,13,24}
      {14}  {12,23}  {12,13,34}
      {23}  {12,24}  {12,14,23}
      {24}  {12,34}  {12,14,34}
      {34}  {13,14}  {12,23,24}
            {13,23}  {12,23,34}
            {13,24}  {12,24,34}
            {13,34}  {13,14,23}
            {14,23}  {13,14,24}
            {14,24}  {13,23,24}
            {14,34}  {13,23,34}
            {23,24}  {13,24,34}
            {23,34}  {14,23,24}
            {24,34}  {14,23,34}
                     {14,24,34}
(End)

B. Bollobas, Modern Graph Theory, Springer, 1998, p. 290.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..388 (first 101 terms from T. D. Noe)
J. E. Bartels, J. Mount, and D. J. A. Welsh, The polytope of win vectors. Annals of Combinatorics 1, 1-15 (1997). https://doi.org/10.1007/BF02558460
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.o
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 132.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 7, arXiv:1709.08416 [math.CO], 2017.
Anton Izosimov, Matrix polynomials, generalized Jacobians, and graphical zonotopes, arXiv:1506.05179 [math.AG], 2015.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See pp. 32-33.
Arun P. Mani and Rebecca J. Stones, Congruences for weighted number of labeled forests, INTEGERS 16 (2016). #A17.
J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
J. Riordan, Letter to N. J. A. Sloane, Oct 19 1970
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1975
John Riordan and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math 6.6 (1980): 333-342.
L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581.
E. M. Wright, A relationship between two sequences, Proc. London Math. Soc. (3) 17 (1967) 296-304.
Index entries for sequences related to forests

The connected case is A000272, rooted A000169.
The unlabeled version is A005195, connected A000055.
The covering case is A105784, unlabeled A144958.
Row sums of A138464.
For triangles instead of cycles we have A213434, covering A372168.
For a unique cycle we have A372193, covering A372195.
A002807 counts cycles in a complete graph.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A006572, A006573, A088956.
Cf. A053530, A054548, A137916, A263340, A322661, A367863, A372176.

exp(x+x^2+add(n^(n-2)*x^n/n!, n=3..50));
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*j^(j-2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 15 2008
# third Maple program:
F:= exp(-LambertW(-x)*(1+LambertW(-x)/2)):
S:= series(F,x,51):
seq(coeff(S,x,j)*j!, j=0..50); # Robert Israel, May 21 2015

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[t-t^2/2],{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2012 *)
nmax = 20; CoefficientList[Series[-LambertW[-x]/(x*E^(LambertW[-x]^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 19 2019 *)

a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n-1,n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!)) /* Michael Somos, Aug 22 2002 */

E.g.f.: exp( Sum_{n>=1} n^(n-2)*x^n/n! ). This implies (by a theorem of Wright) that a(n) ~ exp(1/2)*n^(n-2). - N. J. A. Sloane, May 12 2008 [Corrected by Philippe Flajolet, Aug 17 2008]
E.g.f.: exp(T - T^2/2), where T = T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is Euler's tree function (see A000169). - Len Smiley, Dec 12 2001
Shifts 1 place left under the hyperbinomial transform (cf. A088956). - Paul D. Hanna, Nov 03 2003
a(0) = 1, a(n) = Sum_{j=0..n-1} C(n-1,j) (j+1)^(j-1) a(n-1-j) if n>0. - Alois P. Heinz, Sep 15 2008

More terms from Michael Somos, Aug 22 2002

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

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

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

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

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) = 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

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

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.

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}]

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

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

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

Gus Wiseman, Apr 24 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
  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

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.

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}]

cp's OEIS Frontend

A099198 A bisection of A002807.

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A099201 A bisection of A002807.

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A005195 Number of forests with n unlabeled nodes.

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A001858 Number of forests of trees on n labeled nodes.

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

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

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