A333733 -id:A333733 - cp's OEIS frontend

A257493 Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 21, 24, 1, 1, 1, 5, 55, 282, 120, 1, 1, 1, 6, 120, 2008, 6210, 720, 1, 1, 1, 7, 231, 10147, 153040, 202410, 5040, 1, 1, 1, 8, 406, 40176, 2224955, 20933840, 9135630, 40320, 1, 1, 1, 9, 666, 132724, 22069251, 1047649905, 4662857360, 545007960, 362880, 1

Offset: 0

Alois P. Heinz, Apr 26 2015

Also the number of ordered factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity). A(2,2) = 3: (2*3)^2 = 36 = 4*9 = 6*6 = 9*4.

			Square array A(n,k) begins:
  1,   1,      1,        1,          1,           1,            1, ...
  1,   1,      1,        1,          1,           1,            1, ...
  1,   2,      3,        4,          5,           6,            7, ...
  1,   6,     21,       55,        120,         231,          406, ...
  1,  24,    282,     2008,      10147,       40176,       132724, ...
  1, 120,   6210,   153040,    2224955,    22069251,    164176640, ...
  1, 720, 202410, 20933840, 1047649905, 30767936616, 602351808741, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]

Columns k=0-9 give: A000012, A000142, A000681, A001500, A172806, A172862, A172894, A172919, A172944, A172958.
Rows n=0+1, 2-9 give: A000012, A000027(k+1), A002817(k+1), A001496, A003438, A003439, A008552, A160318, A160319.
Main diagonal gives A110058.
Cf. A257463 (unordered factorizations), A333733 (non-isomorphic matrices), A008300 (binary matrices).

with(numtheory):
b:= proc(n, k) option remember; `if`(n=1, 1, add(
      `if`(bigomega(d)=k, b(n/d, k), 0), d=divisors(n)))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[n_, k_] := b[n, k] = If[n==1, 1, Sum[If[PrimeOmega[d]==k, b[n/d, k], 0], {d, Divisors[n]}]]; A[n_, k_] := b[Product[Prime[i], {i, 1, n}]^k, k]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

T(n, k)={
  local(M=Map(Mat([n, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(k, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, e\m), self()(if(j==t, k, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k, src[i, 1], 0, src[i, 2], k))); vecsum(Mat(M)[, 2])
} \\ Andrew Howroyd, Apr 04 2020

bigomega = sloane.A001222
@cached_function
def b(n, k):
    if n == 1:
        return 1
    return sum(b(n//d, k) if bigomega(d) == k else 0 for d in n.divisors())
def A(n, k):
    return b(prod(nth_prime(i) for i in (1..n))^k, k)
[A(n, d-n) for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018, translated from Maple

from sage.combinat.integer_matrices import IntegerMatrices
[IntegerMatrices([d-n]*n, [d-n]*n).cardinality() for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018

A321721 Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 10, 7, 12, 2, 38, 2, 21, 46, 72, 2, 162, 2, 420, 415, 64, 2, 4987, 1858, 110, 9336, 45456, 2, 136018, 2, 1014658, 406578, 308, 3996977, 34937078, 2, 502, 28010167, 1530292965, 2, 508164038, 2, 54902992348, 51712929897, 1269, 2, 3217847072904, 8597641914, 9168720349613

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d, for some d|n.

			Non-isomorphic representatives of the a(2) = 2 through a(6) = 7 multiset partitions:
  {{11}}   {{111}}     {{1111}}       {{11111}}         {{111111}}
  {{1}{2}} {{1}{2}{3}} {{11}{22}}     {{1}{2}{3}{4}{5}} {{111}{222}}
                       {{12}{12}}                       {{112}{122}}
                       {{1}{2}{3}{4}}                   {{11}{22}{33}}
                                                        {{11}{23}{23}}
                                                        {{12}{13}{23}}
                                                        {{1}{2}{3}{4}{5}{6}}
Inequivalent representatives of the a(6) = 7 matrices:
  [6]
.
  [3 0] [2 1]
  [0 3] [1 2]
.
  [2 0 0] [2 0 0] [1 1 0]
  [0 2 0] [0 1 1] [1 0 1]
  [0 0 2] [0 1 1] [0 1 1]
.
  [1 0 0 0 0 0]
  [0 1 0 0 0 0]
  [0 0 1 0 0 0]
  [0 0 0 1 0 0]
  [0 0 0 0 1 0]
  [0 0 0 0 0 1]
Inequivalent representatives of the a(9) = 7 matrices:
  [9]
.
  [3 0 0] [3 0 0] [2 1 0] [2 1 0] [1 1 1]
  [0 3 0] [0 2 1] [1 1 1] [1 0 2] [1 1 1]
  [0 0 3] [0 1 2] [0 1 2] [0 2 1] [1 1 1]
.
  [1 0 0 0 0 0 0 0 0]
  [0 1 0 0 0 0 0 0 0]
  [0 0 1 0 0 0 0 0 0]
  [0 0 0 1 0 0 0 0 0]
  [0 0 0 0 1 0 0 0 0]
  [0 0 0 0 0 1 0 0 0]
  [0 0 0 0 0 0 1 0 0]
  [0 0 0 0 0 0 0 1 0]
  [0 0 0 0 0 0 0 0 1]

Wikipedia, Magic square

Cf. A006052, A007716, A057150, A120732, A271103, A319056, A319616.

Cf. A321717, A321718, A321719, A321722, A321724, A333733.

a(p) = 2 for p prime corresponding to the 1 X 1 square [p] and the permutation matrices of size p X p with partition (1...10...0). - Chai Wah Wu, Jan 16 2019

a(n) = Sum_{d|n} A333733(d,n/d) for n > 0. - Andrew Howroyd, Apr 11 2020

a(11)-a(13) from Chai Wah Wu, Jan 16 2019
a(14)-a(15) from Chai Wah Wu, Jan 20 2019
Terms a(16) and beyond from Andrew Howroyd, Apr 11 2020

A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 5, 1, 1, 1, 1, 3, 9, 12, 7, 1, 1, 1, 1, 4, 13, 33, 29, 11, 1, 1, 1, 1, 4, 20, 74, 142, 79, 15, 1, 1, 1, 1, 5, 28, 163, 556, 742, 225, 22, 1, 1, 1, 1, 5, 39, 319, 1919, 5369, 4454, 677, 30, 1, 1

Offset: 0

Andrew Howroyd, Apr 08 2020

Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A318805 can be used to extend this method to the unlabeled case.

			Array begins:
==============================================
n\k | 0 1  2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1  1   1    1     1      1       1 ...
  1 | 1 1  1   1    1     1      1       1 ...
  2 | 1 1  2   2    3     3      4       4 ...
  3 | 1 1  3   5    9    13     20      28 ...
  4 | 1 1  5  12   33    74    163     319 ...
  5 | 1 1  7  29  142   556   1919    5793 ...
  6 | 1 1 11  79  742  5369  31781  156191 ...
  7 | 1 1 15 225 4454 64000 692599 5882230 ...
  ...
The T(3,3) = 5 matrices are:
   [0 0 3]  [0 1 2]  [0 1 2]  [1 0 2]  [1 1 1]
   [0 3 0]  [1 1 1]  [1 2 0]  [0 3 0]  [1 1 1]
   [3 0 0]  [2 1 0]  [2 0 1]  [2 0 1]  [1 1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..377
Ming Yean Lim, The number of irreducibles in the plethysm s_lambda[s_m], arXiv:2503.21108 [math.CO], 2025. See p. 8.

Rows n=0..5 are A000012, A000012, A008619, A106607, A333886, A333887.
Columns n=0..5 are A000012, A000012, A000041, A333888, A333889, A333890.
Main diagonal is A333738.
Cf. A188403 (labeled case), A333159 (binary), A333733 (not necessarily symmetric).
Cf. A318805, A333893.

A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 1, 1, 5, 1, 1, 3, 7, 1, 1, 1, 1, 4, 9, 12, 11, 1, 1, 1, 1, 4, 15, 1, 1, 13, 31, 1, 1, 5, 43, 22, 1, 1, 1, 1, 5, 22, 103, 30, 1, 1, 1, 1, 6, 106, 264, 42, 1, 1, 30, 383, 1, 1, 6, 56, 1, 1, 1, 1, 7, 45, 321, 2804, 1731, 77, 1

Offset: 1

Gus Wiseman, Nov 18 2018

Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d.
A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Triangle begins:
  1
  1 1
  1 1
  1 2 1
  1 1
  1 2 3 1
  1 1
  1 3 5 1
  1 5 1
  1 3 7 1
Inequivalent representatives of the T(10,3) = 7 semi-magic squares (zeros not shown):
  [2    ] [2    ] [2    ] [2    ] [2    ] [11   ] [11   ]
  [ 2   ] [ 2   ] [ 2   ] [ 11  ] [ 11  ] [11   ] [1 1  ]
  [  2  ] [  2  ] [  11 ] [ 11  ] [ 1 1 ] [  11 ] [ 1 1 ]
  [   2 ] [   11] [  1 1] [   11] [  1 1] [  1 1] [  1 1]
  [    2] [   11] [   11] [   11] [   11] [   11] [   11]

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

Row sums are A321721.
Cf. A006052, A007016, A007716, A027750, A057150, A120732, A271103, A319056, A319616.
Cf. A321718, A321719, A321722, A333733.

T(n,k) = A333733(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Apr 11 2020

a(28)-a(39) from Chai Wah Wu, Jan 16 2019
Terms a(40) and beyond from Andrew Howroyd, Apr 11 2020
Edited by Peter Munn, Mar 05 2025

A052282 Number of 3 X 3 stochastic matrices under row and column permutations.

Original entry on oeis.org

1, 1, 3, 5, 9, 13, 22, 30, 45, 61, 85, 111, 149, 189, 244, 304, 381, 465, 571, 685, 825, 977, 1158, 1354, 1585, 1833, 2121, 2431, 2785, 3165, 3596, 4056, 4573, 5125, 5739, 6393, 7117, 7885, 8730, 9626, 10605, 11641, 12769, 13959, 15249, 16609, 18076, 19620

Offset: 0

Vladeta Jovovic, Feb 06 2000

nonn

Unreduced numerators in convergent to log(2) = lim[n->inf, a(n)/A000670(n+1)].

			There are 5 nonisomorphic 3 X 3 matrices with row and column sums 3:
[0 0 3] [0 0 3] [0 1 2] [0 1 2] [1 1 1]
[0 3 0] [1 2 0] [1 1 1] [1 2 0] [1 1 1]
[3 0 0] [2 1 0] [2 1 0] [2 0 1] [1 1 1]

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Row n=3 of A333733.

Cf. A002817, A052280, A052281. Different from A001993.

a:= n -> (Matrix([[1, 0, 0, 1, 1, 3, 5, 9, 13]]). Matrix(9, (i,j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -3, -1, 1, 3, -1, -2, 1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2008

LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{1,1,3,5,9,13,22,30,45},50] (* Harvey P. Dale, Mar 10 2018 *)

G.f.: (x^6-x^5+x^3-x+1)/((1-x)^5*(1+x)^2*(1+x+x^2)). - Ralf Stephan and Vladeta Jovovic, May 07 2004

A377060 Array read by antidiagonals: T(n,k) is the number of inequivalent n X k nonnegative integer matrices with all column sums n and row sums k up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 5, 3, 1, 1, 1, 1, 3, 9, 9, 3, 1, 1, 1, 1, 4, 14, 43, 14, 4, 1, 1, 1, 1, 4, 28, 147, 147, 28, 4, 1, 1, 1, 1, 5, 44, 661, 1856, 661, 44, 5, 1, 1, 1, 1, 5, 73, 2649, 25888, 25888, 2649, 73, 5, 1, 1

Offset: 0

Andrew Howroyd, Oct 14 2024

Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A333901. Burnside's lemma can be used to extend this method to the unlabeled case. This seems to require looping over partitions for both rows and columns.

			Array begins:
==================================================
n\k | 0 1 2  3    4      5        6          7 ...
----+---------------------------------------------
  0 | 1 1 1  1    1      1        1          1 ...
  1 | 1 1 1  1    1      1        1          1 ...
  2 | 1 1 2  2    3      3        4          4 ...
  3 | 1 1 2  5    9     14       28         44 ...
  4 | 1 1 3  9   43    147      661       2649 ...
  5 | 1 1 3 14  147   1856    25888     346691 ...
  6 | 1 1 4 28  661  25888  1217727   55138002 ...
  7 | 1 1 4 44 2649 346691 55138002 8597641912 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..324 (first 25 antidiagonals)

Main diagonal is A333734.
Columns k=0..4 are A000012, A000012, A008619, A377061, A377062.
Cf. A333733, A333901, A377007.

T(n,k) = T(k,n).

A052280 Number of 4 X 4 stochastic matrices under row and column permutations.

Original entry on oeis.org

1, 1, 5, 12, 43, 106, 321, 787, 1960, 4354, 9386, 18790, 36362, 66789, 118936, 203840, 340195, 551192, 873343, 1351457, 2052221, 3056798, 4480565, 6462678, 9194098, 12902867, 17892986, 24524478, 33265476, 44666016, 59426834, 78364873, 102502765, 133024660, 171390035, 219278224

Offset: 0

Vladeta Jovovic, Feb 06 2000

nonn

			There are 5 nonisomorphic 4 X 4 matrices with row and column sums 2:
[0 0 0 2] [0 0 0 2] [0 0 0 2] [0 0 1 1] [0 0 1 1]
[0 0 2 0] [0 0 2 0] [0 1 1 0] [0 0 1 1] [0 1 0 1]
[0 2 0 0] [1 1 0 0] [1 0 1 0] [1 1 0 0] [1 0 1 0]
[2 0 0 0] [1 1 0 0] [1 1 0 0] [1 1 0 0] [1 1 0 0]

Row n=4 of A333733.

Cf. A001496, A052281, A052282.

Terms a(9) and beyond from Andrew Howroyd, Apr 04 2020

A232215 Number of n X n matrices (up to permutation of their rows and columns) with nonnegative integer entries with all row and column sums equal to 3.

Original entry on oeis.org

1, 1, 2, 5, 12, 31, 103, 383, 1731, 9273, 57563, 406465, 3212131, 28009976, 266688867, 2749264797, 30480560319, 361435864747, 4562860845767, 61084137737436, 864206301930764, 12882343725953858, 201788397502682460, 3313420771907580764, 56910480298885139055

Offset: 0

N. J. A. Sloane, Nov 22 2013

nonn

Arises from counting of symmetric tensor invariants without color. See Geloun-Ramgoolam, Section 6.2 for information and Mathematica code.

			a(2) = 2 because there are 2 such 2 X 2 matrices: [1 2;2 1] and [3 0;0 3]. - _Nathaniel Johnston_, Oct 12 2016

Mehmet Emin Aktas, Dessins d'Enfants of Trigonal Curves, arXiv:1706.09956 [math.AG], 2017, Theorem 5.
J. B. Geloun, S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
Brendan McKay, Number of all different n-by-n matrices where sum of rows and columns is 3, MathOverflow, 2016.
B. D. McKay and N. C. Wormald, Autormorphisms of Random Graphs with Specified Vertices. Combinatorica 4 (4) (1984) 325-338.

Column k=3 of A333733.

Cf. A328159.

a(n) = 1 + Sum_{i=1..n} A328159(i). - Brendan McKay, Oct 05 2019

New name and a(9)-a(11) from Nathaniel Johnston, Oct 12 2016
a(12) and a(13) from Brendan McKay, Oct 05 2019
a(0)=1 prepended, a(12)-a(13) corrected and terms a(14) and beyond from Andrew Howroyd, Apr 04 2020

A333734 Number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to n up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 2, 5, 43, 1856, 1217727, 8597641912, 646296747486387, 535435113671568180963, 5081029530811947425598907884, 570680215340337514993573217774604779, 779646755088025699677478853259568262608053838

Offset: 0

Andrew Howroyd, Apr 04 2020

			The a(2) = 2 matrices are:
  [1 1]  [2 0]
  [1 1]  [0 2]
.
The a(3) = 5 matrices are:
  [1 1 1]   [2 1 0]   [2 1 0]   [3 0 0]   [3 0 0]
  [1 1 1]   [1 1 1]   [0 2 1]   [0 2 1]   [0 3 0]
  [1 1 1]   [0 1 2]   [1 0 2]   [0 1 2]   [0 0 3]

Main diagonal of A333733 and A377060.

Cf. A110058.

a(11)-a(12) from Andrew Howroyd, Oct 14 2024

A232216 Number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to 4 up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 3, 9, 43, 264, 2804, 44524, 1012456, 30502320, 1166185222, 54902972542, 3115560948081, 209575881782122, 16484822517084705, 1498893370236782629, 155996155818460127750, 18424063865105027559797, 2450806855748517847761175, 364738110721163795332103692, 60368334488355648041244493885

Offset: 0

N. J. A. Sloane, Nov 22 2013

nonn

Arises from counting of symmetric tensor invariants without color. See Geloun-Ramgoolam, Section 6.2 for information and Mathematica code.

Erick Castro, Itzhak Roditi, A combinatorial matrix approach for the generation of vacuum Feynman graphs multiplicities in phi-4 theory, arXiv:1804.08031 [math-ph], 2018.
J. B. Geloun, S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.

Column k=4 of A333733.

Name changed, a(0)=1 prepended and a(9)-a(20) from Andrew Howroyd, Apr 04 2020

cp's OEIS Frontend

A257493 Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

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A321721 Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n.

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A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.

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A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

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A052282 Number of 3 X 3 stochastic matrices under row and column permutations.

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A377060 Array read by antidiagonals: T(n,k) is the number of inequivalent n X k nonnegative integer matrices with all column sums n and row sums k up to permutation of rows and columns.

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A052280 Number of 4 X 4 stochastic matrices under row and column permutations.

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A232215 Number of n X n matrices (up to permutation of their rows and columns) with nonnegative integer entries with all row and column sums equal to 3.

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