A318951 -id:A318951 - 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

A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

Original entry on oeis.org

1, 1, 4, 6, 17, 14, 66, 30, 189, 222, 550, 112, 4696, 202, 5612, 30914, 63219, 594, 453125, 980, 3602695, 5914580, 1169348, 2510, 299083307, 232988061, 23248212, 2669116433, 14829762423, 9130, 170677509317, 13684, 1724710753084, 2199418340875, 14184712185, 38316098104262

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.
Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018
From Gus Wiseman, Oct 11 2018: (Start)
Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{1},{2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{3,4}}
  {{1},{1},{1,1}}
  {{1},{1},{2,2}}
  {{1},{2},{1,2}}
  {{1},{2},{3,4}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{3},{4}}
(End)

			Non-isomorphic representatives of the a(4) = 17 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{2},{2}}
  {{1},{2},{3},{3}}
  {{1},{2},{3},{4}}

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

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951.

Cf. A064573, A279787, A305551, A319616, A319056, A331485.

permcount[v_List] := 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];
K[q_List, 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[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
a[n_] := a[n] = If[n==0, 1, If[PrimeQ[n], 2 PartitionsP[n], Sum[ RowSumMats[ n/d, n, d], {d, Divisors[n]}]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 07 2019, after Andrew Howroyd *)

\\ See A318951 for RowSumMats.
a(n)={sumdiv(n,d,RowSumMats(n/d,n,d))} \\ Andrew Howroyd, Sep 05 2018

For p prime, a(p) = 2*A000041(p).

a(n) = Sum_{d|n} A331485(n/d, d). - Andrew Howroyd, Feb 09 2020

Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018

A219727 Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 9, 3, 1, 1, 15, 66, 31, 5, 1, 1, 52, 712, 686, 109, 7, 1, 1, 203, 10457, 27036, 6721, 339, 11, 1, 1, 877, 198091, 1688360, 911838, 58616, 1043, 15, 1, 1, 4140, 4659138, 154703688, 231575143, 26908756, 476781, 2998, 22, 1

Offset: 0

Alois P. Heinz, Nov 26 2012

A(n,k) is the number of factorizations of m^n where m is a product of k distinct primes. A(2,2) = 9: (2*3)^2 = 36 has 9 factorizations: 36, 3*12, 4*9, 3*3*4, 2*18, 6*6, 2*3*6, 2*2*9, 2*2*3*3.

A(n,k) is the number of (n*k) X k matrices with nonnegative integer entries and column sums n up to permutation of rows. - Andrew Howroyd, Dec 10 2018

			A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)].
A(2,2) = 9: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(2,0),(0,1),(0,1)], [(1,2),(1,0)], [(1,1),(1,1)], [(1,1),(1,0),(0,1)], [(1,0),(1,0),(0,2)], [(1,0),(1,0),(0,1),(0,1)].
Square array A(n,k) begins:
  1,   1,    1,      1,        1,         1,         1,       1, ...
  1,   1,    2,      5,       15,        52,       203,     877, ...
  1,   2,    9,     66,      712,     10457,    198091, 4659138, ...
  1,   3,   31,    686,    27036,   1688360, 154703688, ...
  1,   5,  109,   6721,   911838, 231575143, ...
  1,   7,  339,  58616, 26908756, ...
  1,  11, 1043, 476781, ...
  1,  15, 2998, ...

Andrew Howroyd, Table of n, a(n) for n = 0..209

Columns k=0..3 give: A000012, A000041, A002774, A219678.
Rows n=0..4 give: A000012, A000110, A020555, A322487, A358781.
Main diagonal gives A322488.
Cf. A188392, A219585 (partitions of {n}^k into distinct k-tuples), A256384, A318284, A318951.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p,n,k), [1,n]); s*q[#q-j]))} \\ Andrew Howroyd, Dec 11 2018

A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 3, 1, 1, 5, 23, 21, 5, 1, 1, 7, 79, 162, 66, 7, 1, 1, 11, 274, 1636, 1338, 192, 11, 1, 1, 15, 1003, 19977, 43686, 10585, 565, 15, 1, 1, 22, 3763, 298416, 2142277, 1178221, 82694, 1579, 22, 1, 1, 30, 14723, 5300296, 149056260, 232984145, 30370346, 612700, 4348, 30, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

A(n,k) is the number of non-isomorphic multiset partitions (multisets of multisets) with k parts each of size n.

			Array begins:
============================================================
n\k | 0  1   2     3        4           5              6
----+-------------------------------------------------------
  0 | 1  1   1     1        1           1              1 ...
  1 | 1  1   2     3        5           7             11 ...
  2 | 1  2   7    23       79         274           1003 ...
  3 | 1  3  21   162     1636       19977         298416 ...
  4 | 1  5  66  1338    43686     2142277      149056260 ...
  5 | 1  7 192 10585  1178221   232984145    74676589469 ...
  6 | 1 11 565 82694 30370346 23412296767 33463656939910 ...
  ...
The A(2,2) = 7 matrices are:
  [1 0]  [2 0]  [1 1]  [2 1]  [2 0]  [1 1]  [2 2]
  [1 0]  [0 1]  [1 0]  [0 1]  [0 2]  [1 1]
  [0 1]  [0 1]  [0 1]
  [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..152

Rows n=0..4 are A000012, A000041, A007717, A058194, A331721.
Columns k=0..3 are A000012, A000041, A331722, A331723.
Cf. A219727, A306017, A316674, A318951, A331315, A331461.

\\ See A318951 for RowSumMats
T(n, k)={RowSumMats(k, n*k, n)}
{ for(n=0, 7, for(k=0, 6, print1(T(n, k), ", ")); print) }

A306017(n) = Sum_{d|n} A(n/d, d).

A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 3, 14, 44, 129, 316, 714, 1452, 2775, 4963, 8478, 13838, 21827, 33306, 49504, 71754, 101871, 141807, 194128, 261570, 347633, 456026, 591384, 758596, 963657, 1212861, 1513806, 1874440, 2304225, 2813030, 3412466, 4114608, 4933519

Offset: 0

Vladeta Jovovic, Nov 24 2000

Andrew Howroyd, Table of n, a(n) for n = 0..1000
V. Jovovic, Illustration of initial terms
V. Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 3 of A318951.
Cf. A002817, A052282, A058390, A058391, A058392.
Cf. A001501, A050535, A050913, A058528, A058783, A058784, A058785.

a[n_] := (m = Mod[n, 6]; (n^3 + 9*n^2 + 39*n + 120)*n^3 + Which[m == 0, 12*(23*n^2 + 32*n + 24), m == 1 || m == 5, 249*n^2 + 303*n + 143, m == 2 || m == 4, 4*(69*n^2 + 96*n + 56), m == 3, 3*(83*n^2 + 101*n + 69)])/288; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 12 2011, after Vladeta Jovovic *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(3, 3, n); \\ Andrew Howroyd, Sep 05 2018

a(n) = (1/6)*(C(C(n + 2, 2) + 2, 3) + 3/2*floor((n + 2)/2)*(C(n + 2, 2) - floor((n + 2)/2)) + 3*C(floor((n + 2)/2) + 2, 3) + 2*floor(C(n + 2, 2)/3) + 2*C(C(n + 2, 2) - 3*floor(C(n + 2, 2)/3) + 2, 3)).

Empirical G.f.: -(x^8 + 3*x^7 + 14*x^6 + 12*x^5 + 15*x^4 + 9*x^3 + 5*x^2 + 1) / ((x-1)^7*(x+1)^3*(x^2+x+1)). - Colin Barker, Dec 27 2012

More terms from Marc LeBrun, Dec 11 2000

A058391 Number of 5 X 5 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 7, 198, 5929, 145168, 2459994, 30170387, 282159907, 2114430613, 13190940964, 70598379694, 331820068035, 1395291176641, 5327752138987, 18698405435444, 60922707883197, 185814239933254, 534246250634068, 1456622823771075

Offset: 0

Vladeta Jovovic, Nov 24 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
V. Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 5 of A318951.

Cf. A002620, A058389, A058390, A058392, A003438.

\\ See A318951 for RowSumMats
a(n)=RowSumMats(5, 5, n); \\ Andrew Howroyd, Sep 05 2018

A058390 Number of 4 X 4 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 5, 53, 458, 3411, 19865, 95214, 383714, 1346183, 4202086, 11905966, 31061806, 75533056, 172800689, 374861365, 775978710, 1541027694, 2949003213, 5458806804, 9805626744, 17140511056

Offset: 0

Vladeta Jovovic, Nov 24 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 4 of A318951.

Cf. A002620, A058389, A058391, A058392, A001496, A052280, A052281.

(* See A318951 for RowSumMats *)
a[n_] := RowSumMats[4, 4, n];
a /@ Range[0, 20] (* Jean-François Alcover, Sep 15 2019, after Andrew Howroyd *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(4, 4, n); \\ Andrew Howroyd, Sep 05 2018

A058392 Number of 6 X 6 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 11, 782, 96073, 9283247, 537001197, 19578605324, 487615778173, 8892272235593, 125319645293555, 1423054983691408, 13451239365449764, 108603794657349271, 764673059329865921, 4775254548845993462, 26820549989969591853, 137072193873357150230, 643738505766475169048

Offset: 0

Vladeta Jovovic, Nov 24 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
V. Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 6 of A318951.

Cf. A002620, A058389-A058391, A003439, A005467.

\\ See A318951 for RowSumMats
a(n)=RowSumMats(6, 6, n); \\ Andrew Howroyd, Sep 05 2018

Terms a(15) and beyond from Andrew Howroyd, Sep 05 2018

A322787 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic multiset partitions of a multiset with d = A027750(n, k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 7, 5, 7, 7, 11, 23, 21, 11, 15, 15, 22, 79, 66, 22, 30, 162, 30, 42, 274, 192, 42, 56, 56, 77, 1003, 1636, 1338, 565, 77, 101, 101, 135, 3763, 1579, 135, 176, 19977, 10585, 176, 231, 14723, 43686, 4348, 231, 297, 297, 385, 59663, 298416, 82694, 11582, 385

Offset: 1

Gus Wiseman, Dec 26 2018

			Triangle begins:
   1
   2   2
   3   3
   5   7   5
   7   7
  11  23  21  11
  15  15
  22  79  66  22
  30 162  30
  42 274 192  42
Non-isomorphic representatives of the multiset partitions counted under row 6:
{123456}           {112233}           {111222}           {111111}
{1}{23456}         {1}{12233}         {1}{11222}         {1}{11111}
{12}{3456}         {11}{2233}         {11}{1222}         {11}{1111}
{123}{456}         {112}{233}         {111}{222}         {111}{111}
{1}{2}{3456}       {12}{1233}         {112}{122}         {1}{1}{1111}
{1}{23}{456}       {123}{123}         {12}{1122}         {1}{11}{111}
{12}{34}{56}       {1}{1}{2233}       {1}{1}{1222}       {11}{11}{11}
{1}{2}{3}{456}     {1}{12}{233}       {1}{11}{222}       {1}{1}{1}{111}
{1}{2}{34}{56}     {11}{22}{33}       {11}{12}{22}       {1}{1}{11}{11}
{1}{2}{3}{4}{56}   {11}{23}{23}       {1}{12}{122}       {1}{1}{1}{1}{11}
{1}{2}{3}{4}{5}{6} {1}{2}{1233}       {1}{2}{1122}       {1}{1}{1}{1}{1}{1}
                   {12}{13}{23}       {12}{12}{12}
                   {1}{23}{123}       {2}{11}{122}
                   {2}{11}{233}       {1}{1}{1}{222}
                   {1}{1}{2}{233}     {1}{1}{12}{22}
                   {1}{1}{22}{33}     {1}{1}{2}{122}
                   {1}{1}{23}{23}     {1}{2}{11}{22}
                   {1}{2}{12}{33}     {1}{2}{12}{12}
                   {1}{2}{13}{23}     {1}{1}{1}{2}{22}
                   {1}{2}{3}{123}     {1}{1}{2}{2}{12}
                   {1}{1}{2}{2}{33}   {1}{1}{1}{2}{2}{2}
                   {1}{1}{2}{3}{23}
                   {1}{1}{2}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 1..207 (rows 1..50)

Row sums are A306017. First column is A000041.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306018, A318951, A319190, A319612, A322784, A322785, A322788, A322792.

\\ See A318951 for RowSumMats
row(n)={my(d=divisors(n)); vector(#d, i, RowSumMats(n/d[i], n, d[i]))}
{ for(n=1, 15, print(row(n))) } \\ Andrew Howroyd, Feb 02 2022

Terms a(28) and beyond from Andrew Howroyd, Feb 02 2022

Name edited by Peter Munn, Mar 05 2025

A058194 Number of n-rowed matrices with entries {0,1,2,3} and all row sums 3, up to row and column permutation.

Original entry on oeis.org

1, 3, 21, 162, 1636, 19977, 298416, 5300296, 110219750, 2639842989, 71859429837, 2198244062193, 74860247277672, 2815351714711122, 116130005180284423, 5222901881792429337, 254791333526874348652, 13420798291405599027605, 760201936044714899316798, 46137860613934391781325337, 2990483661567310913388458734

Offset: 0

Vladeta Jovovic, Nov 27 2000

nonn

Row n=3 of A331485.

Cf. A007717, A058407, A058409, A050913.

\\ See A318951 for RowSumMats
a(n)=RowSumMats(n, 3*n, 3); \\ Andrew Howroyd, Sep 06 2018

a(11)-a(20) from Andrew Howroyd, Sep 06 2018

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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A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

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A219727 Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

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A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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A058391 Number of 5 X 5 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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A058390 Number of 4 X 4 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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Mathematica

PARI

A058392 Number of 6 X 6 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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