A339063 -id:A339063 - cp's OEIS frontend

A339065 Number of unlabeled loopless multigraphs with n edges rooted at two noninterchangeable vertices.

1, 4, 17, 69, 281, 1147, 4784, 20345, 88726, 396971, 1823920, 8605364, 41684417, 207201343, 1056244832, 5518054182, 29521703655, 161625956908, 904857279576, 5176569819167, 30241443710950, 180293374961036, 1096240011165724, 6793998104717138, 42894087222036022, 275735424352928682

Offset: 0

Andrew Howroyd, Nov 22 2020

nonn

			The a(1) = 4 cases correspond to a single edge which can be attached to zero, one or both of the roots.

Cf. A050535, A007717 (one root), A339042, A339063, A339066.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[With[{g = GCD[v[[i]], v[[j]]]}, t[v[[i]]*v[[j]]/ g]^g], {i, 2, Length[v]}, {j, 1, i - 1}]*Product[With[{c = v[[i]]}, t[c]^Quotient[c-1, 2]*If[OddQ[c], 1, t[c/2]]], {i, 1, Length[v]}];
G[n_, x_, r_] := Module[{s = 0}, Do[s += permcount[p]*edges[Join[r, p], 1/(1 - x^#) &], {p, IntegerPartitions[n]}]; s/n!];
seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1, 1}]//CoefficientList[#, x]&];
seq[15] (* Jean-François Alcover, Dec 01 2020, after Andrew Howroyd *)

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n, x, r)={my(s=0); forpart(p=n, s+=permcount(p)*edges(concat(r, Vec(p)), i->1/(1-x^i))); s/n!}
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1])))}

A053419 Number of graphs with loops (symmetric relations) with n edges.

Original entry on oeis.org

1, 2, 5, 14, 38, 107, 318, 972, 3111, 10410, 36371, 132656, 504636, 1998361, 8224448, 35112342, 155211522, 709123787, 3342875421, 16234342515, 81102926848, 416244824068, 2192018373522, 11831511359378, 65387590986455, 369661585869273, 2135966349269550, 12604385044890628

Offset: 0

Vladeta Jovovic, Jan 10 2000

nonn

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. a(n) is the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices. For example, non-isomorphic representatives of the a(2) = 5 multiset partitions are (1)(122), (11)(22), (1)(1)(22), (1)(2)(12), (1)(1)(2)(2). - Gus Wiseman, Jul 18 2018

a(n) is the number of unlabeled simple graphs with n edges rooted at one vertex. - Andrew Howroyd, Nov 22 2020

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

Cf. A000664, A000666, A007716, A007717, A020555, A050535, A053419, A094574, A191970 (multisets), A316974, A339063.

seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1}] // CoefficientList[#, x]&]; (* Jean-François Alcover, Dec 03 2020, using Andrew Howroyd's code for g.f. G in A339063 *)

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ Andrew Howroyd, Nov 22 2020

Euler transform of A191970. - Andrew Howroyd, Oct 22 2019

a(16)-a(24) from Max Alekseyev, Jan 22 2010

Terms a(25) and beyond from Andrew Howroyd, Oct 22 2019

A303832 The number of edge-rooted unlabeled connected graphs with n edges.

Original entry on oeis.org

1, 1, 4, 10, 32, 101, 346, 1220, 4517, 17338, 69107, 285009, 1215015, 5344224, 24223641, 113001129, 541913075, 2668817544, 13484234188, 69831773559, 370361639587, 2009988998148, 11153858854425, 63242354288220, 366140089188603, 2163036956456422, 13031489297543608

Offset: 1

R. J. Mathar, May 04 2018

nonn

			a(1)=1: the connected graph with 1 edge (which is rooted).
a(2)=1: the connected graph with 2 edges (one rooted).
a(3)=4: the triangle graph with one choice of rooting, the linear tree with either the middle or a terminating edge rooted, the star graph with one edge rooted.

Cf. A126133 (not necessarily connected), A000664, A303830 (by number of nodes).

Cf. A002905, A339039, A339040, A339041, A339044.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/(2*G(2*n, x+A, [])*(1+x)))} \\ Andrew Howroyd, Nov 21 2020

G.f. A(x) satisfies: A(x)*A000664(x) = A126133(x).

A339040 Number of unlabeled connected simple graphs with n edges rooted at two noninterchangeable vertices.

Original entry on oeis.org

1, 3, 10, 35, 125, 460, 1747, 6830, 27502, 113987, 485971, 2129956, 9591009, 44341610, 210345962, 1023182861, 5100235807, 26035673051, 136023990102, 726877123975, 3970461069738, 22156281667277, 126234185382902, 733899631974167, 4351500789211840

Offset: 1

Andrew Howroyd, Nov 20 2020

nonn

Cf. A000664, A002905, A303832, A304074, A339039, A339041, A339042, A339044, A339063.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A, [])); Vec(G(2*n, x+A, [1, 1])/g - (G(2*n, x+A, [1])/g)^2)}

G.f.: f(x)/g(x) - r(x)^2 where f(x), g(x) and r(x) are the g.f.'s of A339063, A000664 and A339039.

A339041 Number of unlabeled connected simple graphs with n edges rooted at two indistinguishable vertices.

Original entry on oeis.org

1, 2, 7, 21, 73, 255, 946, 3618, 14376, 58957, 249555, 1087828, 4878939, 22488282, 106432530, 516783762, 2572324160, 13116137104, 68461594211, 365559412868, 1995532789212, 11129600885183, 63381069498524, 368338847181336, 2183239817036378

Offset: 1

Andrew Howroyd, Nov 20 2020

nonn

Cf. A000664, A002905, A303831, A303832, A339039, A339040, A339043, A339044, A339063, A339064.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A, []), gr=G(2*n, x+A, [1])/g); Vec(G(2*n, x+A, [1, 1])/g - gr^2 + G(2*n, x+A, [2])/g - subst(gr, x, x^2))/2}

G.f.: f(x)/g(x) - (r(x)^2 + r(x^2))/2 where f(x), g(x) and r(x) are the g.f.'s of A339064, A000664 and A339039.

A339039 Number of unlabeled connected simple graphs with n edges rooted at one distinguished vertex.

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 114, 367, 1248, 4446, 16526, 63914, 256642, 1067388, 4590201, 20376849, 93240065, 439190047, 2126970482, 10579017047, 53983000003, 282345671127, 1512273916781, 8287870474339, 46438619162441, 265840311066579

Offset: 0

Andrew Howroyd, Nov 20 2020

nonn

Cf. A000664, A002905, A053419, A126100, A303832, A339036, A339040, A339041, A339044, A339063.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1])/G(2*n, x+A, []))}

G.f.: f(x)/g(x) where f(x) is the g.f. of A053419 and g(x) is the g.f. of A000664.

A339044 Number of unlabeled connected simple graphs with n edges rooted at one oriented edge.

Original entry on oeis.org

1, 2, 6, 18, 57, 188, 651, 2336, 8719, 33741, 135185, 559908, 2394326, 10557283, 47943126, 223987316, 1075455181, 5301593544, 26807904317, 138924912857, 737220195148, 4002876571636, 22221898966507, 126042573704637, 729944250603862, 4313430995825272

Offset: 1

Andrew Howroyd, Nov 21 2020

nonn

Cf. A000664, A002905, A303832, A304072, A339037, A339039, A339040, A339041, A339063.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1, 1])/G(2*n, x+A, [])/(1+x))}

G.f.: x*f(x)/((1+x)*g(x)) where f(x) is the g.f. of A339063 and g(x) is the g.f. of A000664.

A339064 Number of unlabeled simple graphs with n edges rooted at two indistinguishable vertices.

Original entry on oeis.org

1, 3, 9, 28, 87, 276, 909, 3086, 10879, 39821, 151363, 597062, 2442044, 10342904, 45301072, 204895366, 955661003, 4590214994, 22675644514, 115068710553, 599149303234, 3197694533771, 17475917252052, 97712883807625, 558481251055893, 3260409769087068

Offset: 0

Andrew Howroyd, Nov 22 2020

nonn

			The a(1) = 3 cases correspond to a single edge which can be attached to zero, one or both of the roots.

Cf. A000664, A053419 (one root), A303829, A339041, A339063, A339065.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/2)}

A126133 Number of edge-rooted unlabeled graphs with n edges.

Original entry on oeis.org

1, 2, 7, 21, 66, 210, 699, 2387, 8492, 31329, 120034, 477028, 1965016, 8377888, 36923184, 167972182, 787688821, 3802526173, 18873118341, 96195592212, 502953711022, 2694740822749, 14781176429303, 82931707378322

Offset: 1

Vladeta Jovovic, Mar 07 2007

nonn

			a(3)=7: the triangular graph with one edge rooted. The disconnected graph of the connected linear graph with 3 nodes aside the connected graph with 2 nodes, 2 choices for the root. The three disconnected graphs with 3 graphs on 2 nodes, one of the three with the root. The connected star graph with one edge rooted. The connected linear graph with four nodes, 2 choices for the root. - _R. J. Mathar_, May 03 2018

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.

Max Alekseyev, Table of n, a(n) for n = 1..40

Cf. A000664, A126122, A303832 (connected), A339063, A339064.

\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/(2*(1+x)))} \\ Andrew Howroyd, Nov 21 2020

G.f.: f(x)*x/(1 + x) where f(x) is the g.f. of A339064. - Andrew Howroyd, Nov 22 2020

Terms a(10) onward from Max Alekseyev, May 03 2018

cp's OEIS Frontend

A339065 Number of unlabeled loopless multigraphs with n edges rooted at two noninterchangeable vertices.

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A053419 Number of graphs with loops (symmetric relations) with n edges.

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A303832 The number of edge-rooted unlabeled connected graphs with n edges.

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A339040 Number of unlabeled connected simple graphs with n edges rooted at two noninterchangeable vertices.

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A339041 Number of unlabeled connected simple graphs with n edges rooted at two indistinguishable vertices.

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A339039 Number of unlabeled connected simple graphs with n edges rooted at one distinguished vertex.

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A339044 Number of unlabeled connected simple graphs with n edges rooted at one oriented edge.

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A339064 Number of unlabeled simple graphs with n edges rooted at two indistinguishable vertices.

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A126133 Number of edge-rooted unlabeled graphs with n edges.

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