A350858 -id:A350858 - 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

A350565 a(n) is the minimum permanent of an n X n matrix using the integers 1 to n^2.

Original entry on oeis.org

1, 1, 10, 436, 52197, 13300936, 6192060119

Offset: 0

Stefano Spezia and Hugo Pfoertner, Jan 20 2022

a(7) <= 4755379618016 from the matrix
  [ 1,  2,  3,  4,  5,  6,  7;
    8, 14, 19, 23, 29, 33, 36;
    9, 15, 21, 26, 32, 37, 41;
   10, 16, 22, 28, 35, 40, 44;
   11, 17, 24, 30, 38, 43, 46;
   12, 18, 25, 31, 39, 45, 48;
   13, 20, 27, 34, 42, 47, 49]. - Pontus von Brömssen, Aug 30 2025

			a(2) = 10:
  [1, 3;
   2, 4]
.
a(3) = 436:
  [1, 3, 2;
   4, 8, 6;
   5, 9, 7]
.
a(4) = 52197:
  [1,  2,  4,  3;
   6,  9, 15, 12;
   5,  8, 13, 11;
   7, 10, 16, 14]
.
a(5) = 13300936:
  [16,  8, 24, 21, 12;
   18,  9, 25, 23, 13;
    3,  1,  5,  4,  2;
   14,  6, 20, 17, 10;
   15,  7, 22, 19, 11]
.
a(6) = 6192060119:
  [36, 35, 33, 31, 27,  6;
   11, 10,  9,  8,  7,  1;
   34, 32, 30, 28, 25,  5;
   22, 21, 19, 18, 16,  3;
   29, 26, 24, 23, 20,  4;
   17, 15, 14, 13, 12,  2]

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

Cf. A085000, A350566 (maximum), A350858, A350859, A358486 (elements 0 to n^2-1).

from itertools import permutations
from sympy import Matrix
def A350565(n): return 1 if n == 0 else min(Matrix(n,n,p).per() for p in permutations(range(1,n**2+1))) # Chai Wah Wu, Jan 21 2022

A350859 Maximal 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, 41, 11868, 12124850, 25767879812, 101120963518528

Offset: 0

Stefano Spezia, Jan 19 2022

			a(2) = 41:
   5    2
   3    7
a(3) = 11868:
  23    5    3
   2   13   19
   7   17   11

Cf. A114533, A180128, A350566, A350858 (minimal).

from itertools import permutations
from sympy import Matrix
def A350859(n): return 1 if n == 0 else max(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

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

A364226 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the first n^2 prime numbers.

Original entry on oeis.org

1, 0, 24, 0, 216, 362664

Offset: 1

Stefano Spezia, Jul 15 2023

			The triangle begins:
  1;
  0,  24;
  0, 216, 362664;
  ...

Cf. A180128 (maximal determinant), A088020 (row sums), A350858 (minimal permanent), A350859 (maximal permanent), A364227 (right diagonal).

Cf. A364203 (with the integers in [n^2]).

A364227 a(n) is the number of n X n nonsingular matrices using all the first n^2 prime numbers.

Original entry on oeis.org

1, 24, 362664

Offset: 1

Stefano Spezia, Jul 15 2023

Right diagonal of A364226.
Cf. A180128 (maximal determinant), A350858 (minimal permanent), A350859 (maximal permanent).
Cf. A364206 (with the integers in [n^2]).

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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A350565 a(n) is the minimum permanent of an n X n matrix using the integers 1 to n^2.

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A350859 Maximal permanent of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

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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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A364226 Triangle read by rows: T(n, k) is the number of n X n matrices of rank k using all the first n^2 prime numbers.

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A364227 a(n) is the number of n X n nonsingular matrices using all the first n^2 prime numbers.

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