A377062 -id:A377062 - cp's OEIS frontend

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.

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).

A377061 Number of inequivalent n X 3 nonnegative integer arrays with row sums 3 and column sums n up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 2, 5, 9, 14, 28, 44, 73, 118, 184, 276, 422, 610, 880, 1248, 1742, 2383, 3250, 4341, 5758, 7550, 9808, 12600, 16103, 20352, 25572, 31890, 39519, 48620, 59548, 72432, 87703, 105629, 126642, 151077, 179572, 212428, 250408, 294024, 344052, 401106, 466260, 540110, 623908

Offset: 0

Andrew Howroyd, Oct 14 2024

nonn

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

Column k=3 of A377060.

Cf. A172743, A377062.

cp's OEIS Frontend

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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