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

Author

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

Keywords

• nonn
• hard
• more
• changed

Comments

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

Examples

			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]
		

Links

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

Crossrefs

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

Programs

Python

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