A333734 -id:A333734

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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.

+10
13

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,13

COMMENTS

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.

LINKS

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

EXAMPLE

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 ...

...

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Apr 04 2020

STATUS

approved

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.

+10
5

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,13

COMMENTS

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.

LINKS

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

FORMULA

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

EXAMPLE

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 ...

...

CROSSREFS

Main diagonal is A333734.

Columns k=0..4 are A000012, A000012, A008619, A377061, A377062.

Cf. A333733, A333901, A377007.