A350565 -id:A350565 - cp's OEIS frontend

A350566 a(n) is the maximum permanent of an n X n matrix using the integers 1 to n^2.

1, 1, 14, 947, 161388, 56558003, 36757837732

Offset: 0

Hugo Pfoertner at the suggestion of Stefano Spezia, Jan 21 2022

a(7) >= 38677620556961 corresponding to the matrix
  14, 25, 39,  3, 45,  2, 42
  32, 21, 10, 46,  5, 47,  8
  31, 20,  9, 48,  1, 49,  6
  44, 24, 18, 33, 13, 34, 15
  22, 29, 35, 12, 36, 11, 37
  16, 26, 38,  7, 43,  4, 40
  23, 41, 30, 19, 27, 17, 28 . - Robert Israel, Mar 19 2025
a(7) >= 38677691168324 corresponding to the matrix
   1,  4, 14, 25, 39, 42, 45
   5,  6, 16, 26, 38, 40, 43
  11, 12, 22, 29, 35, 36, 37
  17, 19, 23, 41, 30, 28, 27
  33, 34, 44, 24, 18, 15, 13
  48, 46, 32, 21, 10,  8,  3
  49, 47, 31, 20,  9,  7,  2. - Pontus von Brömssen, Mar 20 2025

			a(2) = 14:
  [2, 3;
   4, 1]
.
a(3) = 947:
  [3, 7, 6;
   9, 4, 1;
   2, 5, 8]
.
a(4) = 161388:
  [ 2,  3, 16,  6;
   11, 13,  4, 10;
    8,  9,  5, 15;
   14, 12,  1,  7]
.
a(5) = 56558003:
  [10,  2, 19, 25,  3;
   11,  5, 23, 20,  8;
   21, 14, 12,  9, 15;
   13, 24,  6,  1, 18;
   16, 17,  7,  4, 22]
.
a(6) = 36757837732:
  [32, 30,  3, 19, 23,  2;
    1,  5, 34, 14, 11, 36;
   17, 18, 15, 31, 22, 16;
   29, 28,  7, 20, 24,  6;
   26, 25, 10, 21, 27,  9;
    4,  8, 35, 13, 12, 33]

Carl-Erik Fröberg, On a combinatorial problem related to permanents, BIT 28 (1988), No. 3, 406-411.

Cf. A085000 (determinant), A350565 (minimum), A350858, A350859, A358487 (elements 0 to n^2-1).

from itertools import permutations
from sympy import Matrix
def A350566(n): return 1 if n == 0 else max(Matrix(n,n,p).per() for p in permutations(range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

A350858 Minimal permanent of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

Original entry on oeis.org

1, 2, 29, 3664, 1820642, 2276752048, 5697057180536

Offset: 0

Stefano Spezia, Jan 19 2022

			a(2) = 29:
    2    3
    5    7
a(3) = 3664:
    2    3    5
    7   13   19
   11   17   23

Cf. A114533, A180128, A350565, A350859 (maximal).

from itertools import permutations
from sympy import Matrix
def A350858(n): return 1 if n == 0 else min(Matrix(n,n,p).per() for p in permutations(prime(m) for m in range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

a(4)-a(6) from Hugo Pfoertner, Jan 21 2022

A358486 a(n) is the minimal permanent of an n X n matrix using the integers 0 to n^2 - 1.

Original entry on oeis.org

1, 0, 2, 128, 18948, 40179728, 2863042492

Offset: 0

Stefano Spezia, Nov 18 2022

			a(3) = 128:
     [0, 1, 2;
      4, 6, 8;
      3, 5, 7]

Cf. A350565 (integers 1 to n^2), A358487 (maximal).

a(4)-a(6) from Hugo Pfoertner, Nov 19 2022

A351610 Minimal permanent of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 7, 163, 9850, 1243806, 284995981

Offset: 0

Stefano Spezia, Feb 14 2022

			a(3) = 163:
   1    2    3
   2    5    4
   3    4    6
a(4) = 9850:
   1    2    3    4
   2    8    5    6
   3    5    9    7
   4    6    7   10

Cf. A000217, A351153, A351611 (maximal).

Cf. A350565, A350858, A350937, A350939, A351019, A351021.

a(5)-a(6) from Hugo Pfoertner, Feb 15 2022

A364203 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the integers from 1 to n^2.

Original entry on oeis.org

1, 0, 24, 0, 2736, 360144

Offset: 1

Stefano Spezia, Jul 13 2023

			The triangle begins:
  1;
  0,   24;
  0, 2736, 360144;
  ...

Cf. A085000 (maximal determinant), A088020 (row sums), A350565 (minimal permanent), A350566 (maximal permanent), A364206 (right diagonal).

Cf. A364226 (with prime numbers).

A364206 a(n) is the number of n X n nonsingular matrices using all the integers from 1 to n^2.

Original entry on oeis.org

1, 24, 360144, 20914499571840

Offset: 1

Stefano Spezia, Jul 13 2023

Right diagonal of A364203.
Cf. A085000 (maximal determinant), A350565 (minimal permanent), A350566 (maximal permanent).
Cf. A364227 (with prime numbers).
Cf. A088020, A136609, A221976.

a(n) = (n^2)! - A221976(n). - Vaclav Kotesovec, Jul 16 2023

a(4) from Vaclav Kotesovec, Jul 16 2023 (using A221976)

cp's OEIS Frontend

A350566 a(n) is the maximum permanent of an n X n matrix using the integers 1 to n^2.

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A350858 Minimal permanent of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

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A358486 a(n) is the minimal permanent of an n X n matrix using the integers 0 to n^2 - 1.

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A351610 Minimal permanent of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

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A364203 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the integers from 1 to n^2.

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A364206 a(n) is the number of n X n nonsingular matrices using all the integers from 1 to n^2.

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