A263542 Triangle T(M, N): Number of M X N matrices where 1

24, 112, 376, 768, 2160, 20352, 5376, 5904, 86208, 51840, 64512, 56736, 1628352, 1342656, 44084736

Offset: 2

Author

Lee Burnette, Oct 20 2015

Keywords

• nonn
• tabl
• more

Comments

This sequence is given in this order: (2,2), (3,2), (3,3), (4,2), (4,3), (4,4), etc.

The idea of the program below is that the first row, first column, and the (1,1)th element uniquely determine the rest of the matrix. Hence, all permutations of m+n integers in the range 0..m*n-1 are generated to fill the first row, first column, and (1,1). Then the empty spots in the matrix are filled in and if at any point a condition is violated (duplicate, < 0, >= M*N), the program immediately moves on to the next permutation.

Much of the conversation in the main chat room of the Programming Puzzles and Code Golf Stack Exchange site - the Nineteenth Byte - following the linked message in the Links section deals with finding the terms of this sequence.

Observation: at least the first 15 terms are divisible by 8. - Omar E. Pol, Oct 20 2015, Nov 21 2015

When M and N are both even, the block sum is 2(MN-1). When one or both is odd the block sum can vary: e.g., for M=N=3, it varies from 12 to 20. - Peter J. Taylor, Oct 21 2015

When M and N are both even, all solutions are toroidal: the block sums created by wrapping from the last column to the first column or the last row to the first row also equal 2(MN-1). When one of M or N is even, all solutions are cylindrical, with wrapping in the even dimension, but they are toroidal only in the trivial case of Odd X 2. When both M and N are odd, except in the trivial case of 1 X 1, solutions do not wrap in either direction. - Peter J. Taylor, Oct 21 2015

Examples

			One 3 X 3 solution (with a sum of 19) is:
   0 4 2
   8 7 6
   3 1 5
One 4 X 4 solution (with a sum of 30) is:
    0  3  4  7
   12 15  8 11
    1  2  5  6
   13 14  9 10
One 5 X 5 solution (with a sum of 48) is:
    0 24  1 23  2
    9 15  8 16  7
   10 14 11 13 12
   19  5 18  6 17
   20  4 21  3 22
The triangle T(M, N) begins:
M\N    2      3       4       5        6 ...
2:    24
3:   112    376
4:   768   2160   20352
5:  5376   5904   86208   51840
6: 64512  56736 1628352 1342656 44084736
...reformatted. - _Wolfdieter Lang_, Dec 16 2015
		

Links

• The Nineteenth Byte, Originating chat message, ChatRoom.

Programs

Python

from itertools import permutations as P
n = 4; m = 4; permutes = P(range(m*n), m+n); counter = 0
for p in permutes:
  grid = [p[:n]]
  for i in range(m-1):
    grid.append([p[n+i]]+[-1]*(n-1))
  grid[1][1] = p[-1]
  s = p[0]+p[1]+p[n]+p[-1]
  has = list(p)
  fail = 0
  for y in range(1,m):
    for x in range(1,n):
      if x == y == 1: continue
      r = s - (grid[y-1][x-1] + grid[y-1][x] + grid[y][x-1])
      if r not in has and 0 <= r < m*n:
        grid[y][x]=r
        has.append(r)
      else:
       fail = 1
       break
    if fail: break
  if not fail:
    counter += 1
print(counter)