A358487 -id:A358487 - 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

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(7) <= 2376296408682 from the matrix
  [ 0,  1,  2,  3,  4,  5,  6;
    7, 13, 19, 23, 29, 34, 37;
    8, 14, 20, 25, 31, 36, 41;
    9, 15, 21, 27, 33, 39, 43;
   10, 16, 22, 28, 35, 42, 45;
   11, 17, 24, 30, 38, 44, 47;
   12, 18, 26, 32, 40, 46, 48]. - Pontus von Brömssen, Sep 03 2025

			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

A358485 a(n) is the maximal determinant of an n X n matrix using the integers 0 to n^2 - 1.

Original entry on oeis.org

1, 0, 6, 332, 36000, 6313388, 1765146660, 731664377274

Offset: 0

Stefano Spezia, Nov 18 2022

427402723914150 <= a(8) <= 427505414757161, 337815614862033534 <= a(9) <= 337888181610225000, 349880703121691699788 <= a(10) <= 349947469107433415221, with upper bounds from corollary 2 of Sigg (2018). - Hugo Pfoertner, Nov 21 2022

			a(3) = 332:
     [5, 7, 2;
      1, 3, 8;
      6, 0, 4]

Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 9 Apr 2018.

Cf. A085000 (integers 1 to n^2), A358486 (minimal permanent), A358487 (maximal permanent).

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

a(7) from Hugo Pfoertner, Nov 21 2022

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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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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A358485 a(n) is the maximal determinant of an n X n matrix using the integers 0 to n^2 - 1.

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