A050532 -id:A050532 - 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

A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 4, 1, 0, 1, 2, 6, 6, 1, 0, 1, 2, 7, 14, 9, 1, 0, 1, 2, 7, 20, 28, 12, 1, 0, 1, 2, 7, 22, 53, 52, 16, 1, 0, 1, 2, 7, 23, 69, 125, 93, 20, 1, 0, 1, 2, 7, 23, 76, 198, 287, 152, 25, 1, 0, 1, 2, 7, 23, 78, 245, 550, 606, 242, 30, 1, 0

Offset: 0

R. J. Mathar, Jul 31 2017

Variant of A138107, here for non-directed edges.

			1   1   1   1    1    1    1   1   1...
0   1   2   2    2    2    2   2   2...
0   1   4   6    7    7    7   7   7...
0   1   6  14   20   22   23  23  23...
0   1   9  28   53   69   76  78  79...
0   1  12  52  125  198  245 264 271...
0   1  16  93  287  550  782 915 973...
0   1  20 152  606 1441 2392
0   1  25 242 1226 3611

Andrew Howroyd, Table of n, a(n) for n = 0..1325
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017. See Table 59.

Cf. A050531 (column 3), A050532 (column 4), A138107, A098568 (vertex-labeled).

rows = 12;
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[g = GCD[v[[i]], v[[j]] ]; t[v[[i]]*v[[j]]/g]^g, {i, 2, Length[v]}, {j, 1, i - 1}]*Product[c = v[[i]]; t[c]^Quotient[c + 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
col[k_] := col[k] = Module[{s = O[x]^rows}, Do[s += permcount[p]*1/edges[p, 1 - x^# + O[x]^rows&], {p, IntegerPartitions[k]}]; s/k!] // CoefficientList[#, x]&;
T[0, ] = 1; T[, 0] = 0;
T[n_, k_] := col[k][[n + 1]];
Table[T[n-k, k], {n, 0, rows-1}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 08 2021, 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)))}
T(m, n=m) = {Mat(vector(n+1, n, my(s=O(x*x^m)); forpart(p=n-1, s+=permcount(p)*1/edges(p,i->1-x^i+O(x*x^m))); Col(s/(n-1)!)))}
{ my(A=T(8)); for(n=1, #A, print(A[n,])) } \\ Andrew Howroyd, Oct 22 2019

A058409 Number of 4 X n nonnegative integer matrices with all column sums 3, up to row and column permutation.

Original entry on oeis.org

1, 3, 18, 92, 458, 1982, 7928, 28592, 95138, 292993, 845027, 2295322, 5915631, 14535676, 34224682, 77510148, 169459010, 358698558, 737106306, 1473867060, 2873564862, 5472731212, 10198198452, 18621343624, 33361303957, 58713103696

Offset: 0

Vladeta Jovovic, Nov 25 2000

Vladeta Jovovic, Number of m x l nonnegative integer matrices with all column sums equal to n, up to row and column permutation

Cf. A050532, A058390, A058407, A058408, A058410.

G.f.: 1 / 24*(1 / (1 - x)^20 + 6 / (1 - x)^6 / (1 - x^2)^7 + 3 / (1 - x^2)^10 + 8 / (1 - x)^2 / (1 - x^3)^6 + 6 / (1 - x^4)^5).

More terms from Max Alekseyev, Jun 21 2011

A058410 Number of 4 X n nonnegative integer matrices with all column sums 4, up to row and column permutation.

Original entry on oeis.org

1, 5, 45, 407, 3411, 25142, 163711, 947909, 4949257, 23579022, 103592423, 423435205, 1622413728, 5864055622, 20102107500, 65660463140, 205176581382, 615506305973, 1778086106680, 4959845850882, 13391444267162, 35072585737475, 89275252614236, 221246652070184

Offset: 0

Vladeta Jovovic, Nov 25 2000

Number of m x l nonnegative integer matrices with all column sums equal to n, up to row and column permutation

Cf. A050532, A058390, A058407-A058409.

G.f.: 1 / 24*(1 / (1 - x)^35 + 6 / (1 - x)^9 / (1 - x^2)^13 + 3 / (1 - x)^3 / (1 - x^2)^16 + 8 / (1 - x)^2 / (1 - x^3)^11 + 6 / (1 - x) / (1 - x^2) / (1 - x^4)^8).

More terms from Sean A. Irvine, Aug 05 2022

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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A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.

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A058409 Number of 4 X n nonnegative integer matrices with all column sums 3, up to row and column permutation.

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Extensions

A058410 Number of 4 X n nonnegative integer matrices with all column sums 4, up to row and column permutation.

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Extensions