A052280 -id:A052280 - cp's OEIS frontend

A333733 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to k up to permutations of rows and columns.

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, 43, 31, 11, 1, 1, 1, 1, 4, 22, 106, 264, 103, 15, 1, 1, 1, 1, 5, 30, 321, 1856, 2804, 383, 22, 1, 1, 1, 1, 5, 45, 787, 12703, 65481, 44524, 1731, 30, 1, 1

Offset: 0

Andrew Howroyd, Apr 04 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 A257493. Burnside's lemma 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        22         30 ...
  4 | 1 1  5  12    43     106       321        787 ...
  5 | 1 1  7  31   264    1856     12703      71457 ...
  6 | 1 1 11 103  2804   65481   1217727   16925049 ...
  7 | 1 1 15 383 44524 3925518 224549073 8597641912 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)

Rows n=0..5 are A000012, A000012, A008619, A052282, A052280, A333735.
Columns k=0..5 are A000012, A000012, A000041, A232215, A232216, A333736.
Main diagonal is A333734.
Cf. A133687, A167625, A257493, A333330, A377060.

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

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

A052281 Number of 4 X 4 symmetric stochastic matrices under row and column permutations.

Original entry on oeis.org

1, 1, 3, 6, 16, 29, 62, 107, 195, 320, 522, 804, 1234, 1804, 2626, 3700, 5155, 7038

Offset: 0

Vladeta Jovovic, Feb 06 2000

This sequence appears to be an erroneous version of A333886.

			There are 6 nonisomorphic symmetric 4 X 4 matrices with row and column sums 3:
[0 0 0 3] [0 0 1 2] [0 0 1 2] [0 0 1 2] [0 0 1 2] [0 1 1 1]
[0 0 3 0] [0 0 2 1] [0 1 1 1] [0 1 2 0] [0 2 1 0] [1 0 1 1]
[0 3 0 0] [1 2 0 0] [1 1 1 0] [1 2 0 0] [1 1 0 1] [1 1 0 1]
[3 0 0 0] [2 1 0 0] [2 1 0 0] [2 0 0 1] [2 0 1 0] [1 1 1 0]
But, A333886 gives 6 other cases.

Cf. A053493, A001496, A052280, A052282.

cp's OEIS Frontend

A333733 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to k up to permutations of rows and columns.

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

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A052281 Number of 4 X 4 symmetric stochastic matrices under row and column permutations.

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