A014068 -id:A014068 - cp's OEIS frontend

A001429 Number of n-node connected unicyclic graphs.

1, 2, 5, 13, 33, 89, 240, 657, 1806, 5026, 13999, 39260, 110381, 311465, 880840, 2497405, 7093751, 20187313, 57537552, 164235501, 469406091, 1343268050, 3848223585, 11035981711, 31679671920, 91021354454, 261741776369, 753265624291, 2169441973139, 6252511838796

Offset: 3

N. J. A. Sloane

Also unlabeled connected simple graphs with n vertices and n edges. The labeled version is A057500. - Gus Wiseman, Feb 12 2024

			From _Gus Wiseman_, Feb 12 2024: (Start)
Representatives of the a(3) = 1 through a(6) = 13 simple graphs:
  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}  {12,13,14,15,16,23}
              {12,13,24,34}  {12,13,14,23,25}  {12,13,14,15,23,26}
                             {12,13,14,23,45}  {12,13,14,15,23,46}
                             {12,13,14,25,35}  {12,13,14,15,26,36}
                             {12,13,24,35,45}  {12,13,14,23,25,36}
                                               {12,13,14,23,25,46}
                                               {12,13,14,23,45,46}
                                               {12,13,14,23,45,56}
                                               {12,13,14,25,26,35}
                                               {12,13,14,25,35,46}
                                               {12,13,14,25,35,56}
                                               {12,13,14,25,36,56}
                                               {12,13,24,35,46,56}
(End)

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
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).

Andrew Howroyd, Table of n, a(n) for n = 3..500
Washington G. Bomfim, A picture of the twenty one unicycles with 3,4,5 and 6 vertices
Audace A. V. Dossou-Olory, Graphs and unicyclic graphs with extremal connected subgraphs, arXiv:1812.02422 [math.CO], 2018.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) Includes illustrations for n <= 6.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
S. Karim, J. Sawada, Z. Alamgirz and S. M. Husnine, Generating bracelets with fixed content, Theoretical Computer Science, Volume 475, 4 March 2013, Pages 103-112.
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Marko Riedel et al., Non-isomorphic, connected, unicyclic graphs, Math Stackexchange, November 2018. (Derivation of algorithm and Maple implementation using PET.)
Marko Riedel, Maple implementation using PET.
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points, Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967. doi: 10.2172/4180737.
Eric Weisstein's World of Mathematics, Unicyclic Graph

For at most one cycle we have A005703, labeled A129271, complement A140637.
Next-to-main diagonal of A054924. Cf. A000055.
The labeled version is A057500, connected case of A137916.
This is the connected case of A137917 and A236570.
Row k = 1 of A137918.
The version with loops is A368983.
A001349 counts unlabeled connected graphs.
A001434 and A006649 count unlabeled graphs with # vertices = # edges.
A006129 counts covering graphs, unlabeled A002494.
Cf. A014068, A054548, A116508, A133686, A140638, A369143, A369201.

Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];Apply[Plus,Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,3,nn}]]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
(* Second program: *)
TreeGf[nn_] := Module[{A}, A = Table[1, {nn}]; For[n = 1, n <= nn 1, n++, A[[n + 1]] = 1/n * Sum[Sum[ d*A[[d]], {d, Divisors[k]}]*A[[n - k + 1]], {k, 1, n}]]; x A.x^Range[0, nn-1]];
seq[n_] := Module[{t, g}, If[n < 3, {}, t = TreeGf[n - 2]; g[e_] := Normal[t + O[x]^(Quotient[n, e]+1)] /. x -> x^e  + O[x]^(n+1); Sum[Sum[ EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[k], g[1]* g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2], {k, 3, n}]]/2 // Drop[CoefficientList[#, x], 3]&];
seq[32] (* Jean-François Alcover, Oct 05 2019, after Andrew Howroyd's PARI code *)

\\ TreeGf gives gf of A000081
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)={if(n<3, [], my(t=TreeGf(n-2)); my(g(e)=subst(t + O(x*x^(n\e)),x,x^e) + O(x*x^n)); Vec(sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2))} \\ Andrew Howroyd, May 05 2018

a(n) = A068051(n) - A027852(n) - A000081(n).

More terms from Ronald C. Read
a(27) corrected, more terms, formula from Christian G. Bower, Feb 12 2002
Edited by Charles R Greathouse IV, Oct 05 2009
Terms a(30) and beyond from Andrew Howroyd, May 05 2018

A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).

Original entry on oeis.org

1, 1, 3, 60, 12600, 37837800, 2053230379200, 2431106898187968000, 73566121315513295589120000, 65191584694745586153436251091200000, 1906765806522767212441719098019963758016000000, 2048024348726152339387799085049745725891853852479488000000

Offset: 0

Clark Kimberling

Number of ways to put numbers 1, 2, ..., n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing. Also number of ways to choose groups of 1, 2, 3, ..., n-1 and n objects out of n*(n+1)/2 objects. - Floor van Lamoen, Jul 16 2001
a(n) is the number of ways to linearly order the multiset {1,2,2,3,3,3,...n,n,...n}. - Geoffrey Critzer, Mar 08 2009
Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

			From _Gus Wiseman_, Aug 12 2020: (Start)
The a(3) = 60 permutations of the prime indices of A006939(3) = 360:
  (111223)  (121123)  (131122)  (212113)  (231211)
  (111232)  (121132)  (131212)  (212131)  (232111)
  (111322)  (121213)  (131221)  (212311)  (311122)
  (112123)  (121231)  (132112)  (213112)  (311212)
  (112132)  (121312)  (132121)  (213121)  (311221)
  (112213)  (121321)  (132211)  (213211)  (312112)
  (112231)  (122113)  (211123)  (221113)  (312121)
  (112312)  (122131)  (211132)  (221131)  (312211)
  (112321)  (122311)  (211213)  (221311)  (321112)
  (113122)  (123112)  (211231)  (223111)  (321121)
  (113212)  (123121)  (211312)  (231112)  (321211)
  (113221)  (123211)  (211321)  (231121)  (322111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..35
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Multinomial Coefficient

Cf. A000178, A014068, A022919, A052295, A208437, A327803.
A190945 counts the case of anti-run permutations.
A317829 counts partitions of this multiset.
A325617 is the version for factorials instead of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A008480 counts permutations of prime indices.
A181818 gives products of superprimorials, with complement A336426.
Cf. A000142, A022559, A027423, A034841, A076954, A112798, A303279, A336417.

with(combinat):
a:= n-> multinomial(binomial(n+1, 2), $0..n):
seq(a(n), n=0..12);  # Alois P. Heinz, May 18 2013

Table[Apply[Multinomial ,Range[n]], {n, 0, 20}]  (* Geoffrey Critzer, Dec 09 2012 *)
Table[Multinomial @@ Range[n], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Binomial[n + 1, 2]!/BarnesG[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Length[Permutations[Join@@Table[i,{i,n},{i}]]],{n,0,4}] (* Gus Wiseman, Aug 12 2020 *)

a(n) = binomial(n+1,2)!/prod(k=1, n, k^(n+1-k)); \\ Michel Marcus, May 02 2019

a(n) = (n*(n+1)/2)!/(0!*1!*2!*...*n!).
a(n) = a(n-1) * A014068(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001.
a(n) = A052295(n)/A000178(n). - Lekraj Beedassy, Feb 19 2004
a(n) = A208437(n*(n+1)/2,n). - Alois P. Heinz, Apr 08 2016
a(n) ~ A * exp(n^2/4 + n + 1/6) * n^(n^2/2 + 7/12) / (2^((n+1)^2/2) * Pi^(n/2)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 02 2019
a(n) = A327803(n*(n+1)/2,n). - Alois P. Heinz, Sep 25 2019
a(n) = A008480(A006939(n)). - Gus Wiseman, Aug 12 2020

More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001
More terms from Michel ten Voorde, Apr 12 2001
Better definition from L. Edson Jeffery, May 18 2013

A137917 a(n) is the number of unlabeled graphs on n nodes whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 14, 35, 97, 264, 733, 2034, 5728, 16101, 45595, 129327, 368093, 1049520, 2999415, 8584857, 24612114, 70652441, 203075740, 584339171, 1683151508, 4852736072, 14003298194, 40441136815, 116880901512, 338040071375, 978314772989, 2833067885748, 8208952443400

Offset: 0

Washington Bomfim, Feb 24 2008

nonn

a(n) is the number of simple unlabeled graphs on n nodes whose components have exactly one cycle. - Geoffrey Critzer, Oct 12 2012

Also the number of unlabeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 5 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
F. Ruskey, Combinatorial Generation, p. 79, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

Cf. A008483, A137918.
The connected case is A001429.
Without the choice condition we have A001434, covering A006649.
For any number of edges we have A134964, complement A140637.
The labeled version is A137916.
The version with loops is A369145, complement A368835.
The complement is counted by A369201, labeled A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A007716, A014068, A053530, A116508, A133686, A140638, A368601, A369141, A369146.

Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,3,nn}]],1];CoefficientList[Series[Product[1/(1-x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x]   (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n],{2}],{n}],Select[Tuples[#],UnsameQ@@#&]!={}&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

a(n) = Sum_{1*j_1 + 2*j_2 + ... = n} (Product_{i=3..n} binomial(A001429(i) + j_i -1, j_i)). [F. Ruskey p. 79, (4.27) with n replaced by n+1, and a_i replaced by A001429(i)].

Euler transform of A001429. - Geoffrey Critzer, Oct 12 2012

Edited by Washington Bomfim, Jun 27 2012
Terms a(30) and beyond from Andrew Howroyd, May 05 2018
Offset changed to 0 by Gus Wiseman, Jan 27 2024

A368597 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges.

Original entry on oeis.org

1, 1, 3, 17, 150, 1803, 27364, 501015, 10736010, 263461265, 7283725704, 223967628066, 7581128184175, 280103206674480, 11216492736563655, 483875783716549277, 22371631078155742764, 1103548801569848115255, 57849356643299101021960, 3211439288584038922502820

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

			The a(0) = 1 through a(3) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,3}}
             {{2},{1,2}}  {{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
Eric Weisstein's World of Mathematics, Graph Loop.

This is the covering case of A014068.
Allowing edges of any positive size gives A054780, covering case of A136556.
Allowing any number of edges gives A322661, connected A062740.
The case of just pairs is A367863, covering case of A116508.
The unlabeled version is A368599.
The version contradicting strict AOC is A368730.
The connected case is A368951.
A000085 counts set partitions into singletons or pairs.
A006129 counts covering graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A000272, A000666, A057500, A066383, A333331, A367869, A368596, A368600, A368928, A368951, A369199.

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

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n)) \\ Andrew Howroyd, Jan 06 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n). - Andrew Howroyd, Jan 06 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 06 2024

A368927 Number of labeled loop-graphs covering a subset of {1..n} such that it is possible to choose a different vertex from each edge.

Original entry on oeis.org

1, 2, 7, 39, 314, 3374, 45630, 744917, 14245978, 312182262, 7708544246, 211688132465, 6397720048692, 210975024924386, 7537162523676076, 289952739051570639, 11949100971787370300, 525142845422124145682, 24515591201199758681892, 1211486045654016217202663

Offset: 0

Gus Wiseman, Jan 15 2024

nonn

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

			The a(0) = 1 through a(2) = 7 loop-graphs (loops shown as singletons):
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

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

Without the choice condition we have A006125.
The case of a unique choice is A088957, unlabeled A087803.
The case without loops is A133686, complement A367867, covering A367869.
For exactly n edges and no loops we have A137916, unlabeled A137917.
For exactly n edges we have A333331 (maybe), complement A368596.
For edges of any positive size we have A367902, complement A367903.
The covering case is A369140, complement A369142.
The complement is counted by A369141.
The complement for n edges and no loops is A369143, covering A369144.
The unlabeled version is A369145, complement A369146.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A000169, A000272, A000666, A014068, A057500, A116508, A367863, A368601, A368924, A368984, A369200.

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

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

Binomial transform of A369140.
Exponential transform of A369197 with A369197(1) = 2.
E.g.f.: 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(7) onwards from Andrew Howroyd, Feb 02 2024

A333331 Number of integer points in the convex hull in R^n of parking functions of length n.

Original entry on oeis.org

1, 3, 17, 144, 1623, 22804, 383415, 7501422

Offset: 1

Richard Stanley, Mar 15 2020

It is observed by Gus Wiseman in A368596 and A368730 that this sequence appears to be the complement of those sequences. If this is the case, then a(n) is the number of labeled graphs with loops allowed in which each connected component has an equal number of vertices and edges and the conjectured formula holds. Terms for n >= 9 are expected to be 167341283, 4191140394, 116425416531, ... - Andrew Howroyd, Jan 10 2024
From Gus Wiseman, Mar 22 2024: (Start)
An equivalent conjecture is that a(n) is the number of loop-graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. I call these graphs choosable. For example, the a(3) = 17 choosable loop-graphs are the following (loops shown as singletons):
  {{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
                 {{1},{2},{2,3}}  {{1},{1,2},{2,3}}
                 {{1},{3},{1,2}}  {{1},{1,3},{2,3}}
                 {{1},{3},{2,3}}  {{2},{1,2},{1,3}}
                 {{2},{3},{1,2}}  {{2},{1,2},{2,3}}
                 {{2},{3},{1,3}}  {{2},{1,3},{2,3}}
                                  {{3},{1,2},{1,3}}
                                  {{3},{1,2},{2,3}}
                                  {{3},{1,3},{2,3}}
(End)

			For n=2 the parking functions are (1,1), (1,2), (2,1). They are the only integer points in their convex hull. For n=3, in addition to the 16 parking functions, there is the additional point (2,2,2).

R. P. Stanley (Proposer), Problem 12191, Amer. Math. Monthly, 127:6 (2020), 563.

Thomas Selig, The stochastic sandpile model on complete graphs, arXiv:2209.07301 [math.PR], 2022.

All of the following relative references pertain to the conjecture:
The case of unique choice A000272.
The version without the choice condition is A014068, covering A368597.
The case of just pairs A137916.
For any number of edges of any positive size we have A367902.
The complement A368596, covering A368730.
Allowing edges of any positive size gives A368601, complement A368600.
Counting by singletons gives A368924.
For any number of edges we have A368927, complement A369141.
The connected case is A368951.
The unlabeled version is A368984, complement A368835.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000169, A000666, A057500, A062740, A129271, A133686, A367863, A367903, A369146, A369199.

Conjectured e.g.f.: exp(-log(1-T(x))/2 + T(x)/2 - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 10 2024

A368596 Number of n-element sets of singletons or pairs of distinct elements of {1..n}, or loop graphs with n edges, such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 3, 66, 1380, 31460, 800625, 22758918, 718821852, 25057509036, 957657379437, 39878893266795, 1799220308202603, 87502582432459584, 4566246347310609247, 254625879822078742956, 15115640124974801925030, 952050565540607423524658, 63425827673509972464868323

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(3) = 3 set-systems:
  {{1},{2},{1,2}}
  {{1},{3},{1,3}}
  {{2},{3},{2,3}}

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

The version without the choice condition is A014068, covering A368597.
The complement appears to be A333331.
For covering pairs we have A367868.
Allowing edges of any positive size gives A368600, any length A367903.
The covering case is A368730.
The unlabeled version is A368835.
A000085 counts set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
A322661 counts covering half-loop-graphs, connected A062740.
A369141 counts non-choosable loop-graphs, covering A369142.
A369146 counts unlabeled non-choosable loop-graphs, covering A369147.
Cf. A000272, A000666, A057500, A129271, A133686, A367769, A367863, A367867, A367869, A367901, A367907, A368097, A369199.

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

Terms a(7) and beyond from Andrew Howroyd, Jan 10 2024

A066383 a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).

Original entry on oeis.org

1, 2, 7, 42, 386, 4944, 82160, 1683218, 40999516, 1156626990, 37060382822, 1328700402564, 52676695500313, 2287415069586304, 107943308165833912, 5499354613856855290, 300788453960472434648, 17577197510240126035698, 1092833166733915284972350

Offset: 0

N. J. A. Sloane, Dec 23 2001

nonn

Number of labeled loop-graphs with n vertices and at most n edges. - Gus Wiseman, Feb 14 2024

			From _Gus Wiseman_, Feb 14 2024: (Start)
The a(0) = 1 through a(2) = 7 loop-graphs (loops shown as singletons):
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100

The case of equality is A014068, covering A368597.
The loopless version is A369192, covering A369191.
The covering case is A369194, minimal case A001862.
Counting only covered vertices gives A369196, without loops A369193.
The connected covering case is A369197, without loops A129271.
The unlabeled version is A370168, covering A370169.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000085, A000666, A005703, A062740, A116508, A367862, A367863, A367916, A368927, A369141, A369199.

f[n_] := Sum[Binomial[n (n + 1)/2, k], {k, 0, n}]; Array[f, 21, 0] (* Vincenzo Librandi, May 06 2016 *)
Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Length[#]<=n&]],{n,0,5}] (* Gus Wiseman, Feb 14 2024 *)

{ for (n=0, 100, a=0; for (k=0, n, a+=binomial(n*(n + 1)/2, k)); write("b066383.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010

from math import comb
def A066383(n): return sum(comb(comb(n+1,2),k) for k in range(n+1)) # Chai Wah Wu, Jul 10 2024

a(n) = 2^(n*(n+1)/2) - binomial(n*(n+1)/2,n+1)*2F1(1,(-n^2+n+2)/2;n+2;-1) = A006125(n) - A116508(n+1) * 2F1(1,(-n^2+n+2)2;n+2;-1), where 2F1(a,b;c;x) is the hypergeometric function. - Ilya Gutkovskiy, May 06 2016

a(n) ~ exp(n) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, Feb 20 2024

A001434 Number of graphs with n nodes and n edges.

Original entry on oeis.org

1, 0, 0, 1, 2, 6, 21, 65, 221, 771, 2769, 10250, 39243, 154658, 628635, 2632420, 11353457, 50411413, 230341716, 1082481189, 5228952960, 25945377057, 132140242356, 690238318754, 3694876952577, 20252697246580, 113578669178222, 651178533855913, 3813856010041981

Offset: 0

N. J. A. Sloane

The labeled version is A116508. - Gus Wiseman, Feb 22 2024

			From _Gus Wiseman_, Feb 22 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 6 graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,24}
                                       {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
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).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 1..40 from Sean A. Irvine)
CombOS - Combinatorial Object Server, Generate graphs
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967.

The connected case is A001429, labeled A057500.
The covering case is A006649, labeled A367863.
Diagonal n = k of A008406.
The labeled version is A116508.
The version with loops is A368598, connected A368983.
Allowing up to n edges gives A370315, labeled A369192.
A000088 counts unlabeled graphs, labeled A006125.
A001349 counts unlabeled connected graphs, labeled A001187.
A002494 counts unlabeled covering graphs, labeled A006129.
Cf. A000055, A005703, A014068, A054924, A137916, A137917, A236570, A369201.

(* first do *) Needs["Combinatorica`"] (* then *) Table[ NumberOfGraphs[n, n], {n, 24}] (* Robert G. Wilson v, Mar 22 2011 *)
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{2}],{n}]]],{n,0,5}] (* Gus Wiseman, Feb 22 2024 *)

a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A008406. - Andrew Howroyd, Feb 02 2024

More terms from Vladeta Jovovic, Jan 07 2000

a(0)=1 prepended by Andrew Howroyd, Feb 02 2024

A368984 Number of graphs with loops (symmetric relations) on n unlabeled vertices in which each connected component has an equal number of vertices and edges.

Original entry on oeis.org

1, 1, 2, 5, 12, 29, 75, 191, 504, 1339, 3610, 9800, 26881, 74118, 205706, 573514, 1606107, 4513830, 12727944, 35989960, 102026638, 289877828, 825273050, 2353794251, 6724468631, 19239746730, 55123700591, 158133959239, 454168562921, 1305796834570, 3758088009136

Offset: 0

Andrew Howroyd, Jan 11 2024

nonn

The graphs considered here can have loops but not parallel edges.

Also the number of unlabeled loop-graphs with n edges and n vertices such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

			Representatives of the a(3) = 5 graphs are:
   {{1,2}, {1,3}, {2,3}},
   {{1}, {1,2}, {1,3}},
   {{1}, {1,2}, {2,3}},
   {{1}, {2}, {2,3}},
   {{1}, {2}, {3}}.
The graph with 4 vertices and edges {{1}, {2}, {1,2}, {3,4}} is included by A368599 but not by this sequence.

Andrew Howroyd, Table of n, a(n) for n = 0..500

The case of a unique choice is A000081.
Without loops we have A137917, labeled A137916.
The labeled version appears to be A333331.
Without the choice condition we have A368598, covering A368599.
The complement is counted by A368835, labeled A368596 (covering A368730).
Row sums of A368926, labeled A368924.
The connected case is A368983.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A014068, A057500, A116508, A129271, A133686, A367863, A367869, A367902, A368597, A368601, A368836.

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n],{1,2}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

Euler transform of A368983.

cp's OEIS Frontend

A001429 Number of n-node connected unicyclic graphs.

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A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!*1!*2!*...*n!).

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A137917 a(n) is the number of unlabeled graphs on n nodes whose components are unicyclic graphs.

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A368597 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges.

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A368927 Number of labeled loop-graphs covering a subset of {1..n} such that it is possible to choose a different vertex from each edge.

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A333331 Number of integer points in the convex hull in R^n of parking functions of length n.

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A368596 Number of n-element sets of singletons or pairs of distinct elements of {1..n}, or loop graphs with n edges, such that it is not possible to choose a different element from each.

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A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).