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

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

Offset: 0

Author

Stefano Spezia and Hugo Pfoertner, Jan 20 2022

Keywords

• nonn
• hard
• more
• changed

Comments

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

Examples

			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]
		

Links

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

Crossrefs

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

Programs

Python

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