A339065 -id:A339065 - cp's OEIS frontend

A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

1, 2, 7, 23, 79, 274, 1003, 3763, 14723, 59663, 250738, 1090608, 4905430, 22777420, 109040012, 537401702, 2723210617, 14170838544, 75639280146, 413692111521, 2316122210804, 13261980807830, 77598959094772, 463626704130058, 2826406013488180, 17569700716557737

Offset: 0

Colin Mallows

nonn

Euler transform of A007719.
Also the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018
Number of distinct n X 2n matrices with integer entries and rows sums 2, up to row and column permutations. - Andrew Howroyd, Sep 06 2018
a(n) is the number of unlabeled loopless multigraphs with n edges rooted at one vertex. - Andrew Howroyd, Nov 22 2020

			a(2) = 7 (here - denotes an edge, = denotes a pair of parallel edges and o is a loop):
  oo
  o o
  o-
  o -
  =
  --
  - -
From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(2) = 7 multiset partitions of {1, 1, 2, 2}:
  (1122),
  (1)(122), (11)(22), (12)(12),
  (1)(1)(22), (1)(2)(12),
  (1)(1)(2)(2).
(End)
From _Gus Wiseman_, Jan 08 2024: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(3) = 7 rooted loopless multigraphs (root shown as singleton):
  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,2}}
         {{1},{2,3}}  {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{1,2},{3,4}}
                      {{1},{2,3},{2,3}}
                      {{1},{2,3},{2,4}}
                      {{1},{2,3},{4,5}}
(End)

Huaien Li and David C. Torney, Enumerations of Multigraphs, 2002.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Mateo Díaz, Dmitriy Drusvyatskiy, Jack Kendrick, and Rekha R. Thomas, Invariant Kernels: Rank Stabilization and Generalization Across Dimensions, arXiv:2502.01886 [math.OC], 2025. See p. 43.
Huaien Li and David C. Torney, Enumeration of unlabelled multigraphs, Ars Combin. 75 (2005) 171-188. MR2133219.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) table 67.

Row n=2 of A331485.

Cf. A000664, A002620, A007716, A007719, A020555, A050531, A050532, A050535, A052171, A053418, A053419, A094574, A316972, A316974, A318951, A339065.

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];
Kq[q_, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s=0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[Kq[q, t, k]/t x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
a[n_] := RowSumMats[n, 2n, 2];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(n, 2*n, 2); \\ Andrew Howroyd, Sep 06 2018

\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ Andrew Howroyd, Nov 22 2020

More terms from Vladeta Jovovic, Jan 26 2000

a(0)=1 prepended and a(16)-a(25) added by Max Alekseyev, Jun 21 2011

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

Original entry on oeis.org

1, 1, 3, 8, 23, 66, 212, 686, 2389, 8682, 33160, 132277, 550835, 2384411, 10709827, 49782637, 238998910, 1182772364, 6023860266, 31525780044, 169316000494, 932078457785, 5253664040426, 30290320077851, 178480713438362, 1073918172017297

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017
Row sums of the triangle defined by the Multiset Transformation of A076864,
;
1;
2 1;
5 2 1;
12 8 2 1;
33 22 8 2 1;
103 72 26 8 2 1;
333 229 87 26 8 2 1;
1183 782 295 92 26 8 2 1;
4442 2760 1036 315 92 26 8 2 1;
17576 10270 3735 1129 321 92 26 8 2 1;
72810 39770 13976 4117 1154 321 92 26 8 2 1;
314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;
- R. J. Mathar, Jul 18 2017
Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123)
  (12)(13)(23)
  (1)(1)(23)(23)
  (1)(2)(3)(123)
  (1)(2)(13)(23)
  (1)(1)(2)(3)(23)
  (1)(1)(2)(2)(3)(3)
(End)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).

Andrew Howroyd, Table of n, a(n) for n = 0..50
George Barnes, Sanjaye Ramgoolam, and Michael Stephanou, Permutation invariant Gaussian matrix models for financial correlation matrices, arXiv:2306.04569 [q-fin.ST], 2023.
Frank Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation
Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, Energy flow polynomials: A complete linear basis for jet substructure, arXiv:1712.07124 [hep-ph], 2017.
Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, Universal graph series, chromatic functions, and their index theory, arXiv:2403.09985 [math.CO], 2024. See p. 23.

Cf. A001399, A003082, A014395, A014396, A014397, A014398.
Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.
Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.

seq[n_] := G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - Max Alekseyev, Oct 18 2017

Euler transform of A076864. - Andrew Howroyd, Oct 23 2019

More terms from Sean A. Irvine, Oct 02 2011

A339063 Number of unlabeled simple graphs with n edges rooted at two noninterchangeable vertices.

Original entry on oeis.org

1, 4, 13, 43, 141, 467, 1588, 5544, 19966, 74344, 286395, 1141611, 4707358, 20063872, 88312177, 400980431, 1875954361, 9032585846, 44709095467, 227245218669, 1184822316447, 6330552351751, 34630331194626, 193785391735685, 1108363501628097, 6474568765976164

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. A000664, A053419 (one root), A304070, A339040, A339064, A339065.

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, 2, Length[v]}];
G[n_, x_, r_] := Module[{s = 0}, Do[s += permcount[p]*edges[Join[r, p], 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 03 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+x^i)); s/n!}
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1])))}

A339036 Number of unlabeled connected loopless multigraphs with n edges rooted at one distinguished vertex.

Original entry on oeis.org

1, 1, 3, 9, 30, 104, 390, 1518, 6208, 26372, 116221, 529341, 2487054, 12027502, 59778867, 304916272, 1594273763, 8535706749, 46753269749, 261771468438, 1497087288210, 8739579074131, 52045067963540, 315980654042243, 1954770128712348, 12315770916526091

Offset: 0

Andrew Howroyd, Nov 20 2020

nonn

Cf. A007717, A050535, A076864, A339037, A339038, A339039, A339042, A339043, A339065.

seq[n_] := G[2n, x+O[x]^n, {1}]/G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

\\ See A339065 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 A007717 and g(x) is the g.f. of A050535.

A339037 Number of unlabeled connected loopless multigraphs with n edges rooted at one oriented edge.

Original entry on oeis.org

1, 3, 11, 41, 160, 641, 2672, 11479, 50938, 232830, 1095151, 5292990, 26257328, 133548307, 695752146, 3709509938, 20224607541, 112675185837, 641016837378, 3721624588590, 22037618432547, 133023405207408, 818085097509494, 5123460267381837, 32660335570381961, 211825198708110059

Offset: 1

Andrew Howroyd, Nov 20 2020

nonn

Cf. A050535, A076864, A339036, A339038, A339042, A339043, A339044, A339065.

seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1, 1}]/G[2n, x+A, {}] // CoefficientList[#, x]&]; (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)

\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1,1])/G(2*n, x+A, []))}

G.f.: x*f(x)/g(x) where f(x) is the g.f. of A339065 and g(x) is the g.f. of A050535.

A339042 Number of unlabeled connected loopless multigraphs with n edges rooted at two noninterchangeable vertices.

Original entry on oeis.org

1, 4, 17, 73, 319, 1423, 6499, 30374, 145302, 711177, 3559690, 18212192, 95193547, 508083746, 2767835600, 15382476029, 87177582535, 503610832756, 2964300557548, 17771210411578, 108471258414870, 673836620069035, 4258727230198033, 27373904651169023, 178885471934461869

Offset: 1

Andrew Howroyd, Nov 20 2020

nonn

Cf. A050535, A076864, A339036, A339037, A339038, A339040, A339043, A339065.

seq[n_] := Module[{g}, g = G[2n, x+O[x]^n, {}]; G[2n, x+O[x]^n, {1, 1}]/g - (G[2n, x+O[x]^n, {1}]/g)^2 // CoefficientList[#, x]& // Rest];
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

\\ See A339065 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)^2 where x*f(x) is the g.f. of A339037 and g(x) is the g.f. of A339036.

A339043 Number of unlabeled connected loopless multigraphs with n edges rooted at two indistinguishable vertices.

Original entry on oeis.org

1, 3, 11, 43, 178, 767, 3425, 15783, 74775, 363639, 1811808, 9239430, 48175945, 256658465, 1396152633, 7750325528, 43882706171, 253308596926, 1490040961732, 8928063141435, 54469529215562, 338236254603888, 2136952452531537, 13731571816349732, 89710429044324926

Offset: 1

Andrew Howroyd, Nov 20 2020

nonn

Cf. A050535, A076864, A339036, A339037, A339038, A339041, A339042, A339065.

seq[n_] := Module[{g, gr}, g = G[2n, x+O[x]^n, {}]; gr = G[2n, x+O[x]^n, {1}]/g; G[2n, x+O[x]^n, {1, 1}]/g - gr^2 + G[2n, x+O[x]^n, {2}]/g - (Normal[gr] /. x -> x^2) // CoefficientList[#/2, x]& // Rest];
seq[15] (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)

\\ See A339065 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(g) - (g(x)^2 + g(x^2))/2 where x*f(x) is the g.f. of A339038 and g(x) is the g.f. of A339036.

A338999 Number of connected multigraphs with n edges and rooted at two indistinguishable vertices whose removal leaves a connected graph.

Original entry on oeis.org

1, 1, 3, 11, 43, 180, 804, 3763, 18331, 92330, 478795, 2547885, 13880832, 77284220, 439146427, 2543931619, 15010717722, 90154755356, 550817917537, 3421683388385, 21601986281226, 138548772267326, 902439162209914, 5967669851051612, 40053432076016812

Offset: 1

Rainer Rosenthal, Nov 18 2020

nonn

This sequence counts the CDE-descendants of a single edge A-Z.
[C]onnect: different nodes {P,Q} != {A,Z} may form a new edge P-Q.
[D]issect: any edge P-Q may be dissected into P-M-Q with a new node M.
[E]xtend:  any node P not in {A,Z} may form a new edge P-Q with a new node Q.
These basic operations were motivated by A338487, which seemed to count the CDE-descendants of K_4 with edge A-Z removed.

			The a(3) = 3 CDE-descendants of A-Z with 3 edges are
.
         A          A          A
        ( )        /          /
         o        o - o      o - o
         |           /        \
         Z          Z          Z
.
        DCC        DD         DE
.

Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.

Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23, 2004.

Cf. A180414, A337517, A338487, A339038, A339045, A339065.

\\ See A339065 for G.
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A,[]), gr=G(2*n, x+A,[1])/g, u=InvEulerT(Vec(-1+G(2*n, x+A,[1,1])/(g*gr^2))), t=InvEulerT(Vec(-1+G(2*n, x+A,[2])/(g*subst(gr,x,x^2)))), v=vector(n)); for(n=1, #v, v[n]=(u[n]+t[n]-if(n%2==0,u[n/2]-v[n/2]))/2); v} \\ Andrew Howroyd, Nov 20 2020

a(7)-a(25) from Andrew Howroyd, Nov 20 2020

A339038 Number of unlabeled connected loopless multigraphs with n edges rooted at one unoriented edge.

Original entry on oeis.org

1, 2, 7, 23, 88, 339, 1396, 5915, 26080, 118539, 555678, 2678458, 13262193, 67353325, 350493424, 1866989802, 10171394388, 56631507822, 322011612423, 1868702977253, 11061267210030, 66745602611831, 410360493588788, 2569318971123439, 16374787277199728, 106180292431149021

Offset: 1

Andrew Howroyd, Nov 20 2020

nonn

Cf. A050535, A076864, A303832, A339036, A339037, A339042, A339043, A339065, A339066.

seq[n_] := (G[2n, x + O[x]^n, {1, 1}] + G[2n, x + O[x]^n, {2}])/G[2n, x + O[x]^n, {}] // CoefficientList[#/2, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)

\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1,1]) + G(2*n, x+A, [2]))/G(2*n, x+A, []))/2}

G.f.: x*f(x)/g(x) where f(x) is the g.f. of A339066 and g(x) is the g.f. of A050535.

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)}

cp's OEIS Frontend

A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

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A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

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A339063 Number of unlabeled simple graphs with n edges rooted at two noninterchangeable vertices.

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A339036 Number of unlabeled connected loopless multigraphs with n edges rooted at one distinguished vertex.

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A339037 Number of unlabeled connected loopless multigraphs with n edges rooted at one oriented edge.

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A339042 Number of unlabeled connected loopless multigraphs with n edges rooted at two noninterchangeable vertices.

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A339043 Number of unlabeled connected loopless multigraphs with n edges rooted at two indistinguishable vertices.

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A338999 Number of connected multigraphs with n edges and rooted at two indistinguishable vertices whose removal leaves a connected graph.

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