A002047 -id:A002047 - cp's OEIS frontend

A375800 Number of ways of placing 2n nonattacking rooks on a hexagonal board of equilateral triangular spaces with n spaces along each edge.

3, 24, 348, 7812, 255756, 11747504, 714121392

Offset: 1

Hugh Robinson, Aug 29 2024

A rook move on the equilateral triangle tessellation is a move along a path through successively edge-adjacent faces, that turns alternately left and right at each face (starting with either left or right), so that the overall direction of movement remains approximately parallel to one set of triangle edges.

Also counts the number of 3 X 2n matrices such that each row is a permutation of {1, .., 2n}, the first row is the identity permutation (1 .. 2n), and each column sums to either 3n+1 or 3n+2. This parallels how the equivalent problem for the board tessellated with hexagons (A002047) counts the number of 3 X (2n-1) zero-sum arrays.

			For n = 2, the a(2) = 24 arrangements are rotations and reflections of:
      o---o---o           o---o---o           o---o---o
     /X\ / \ / \         /X\ / \ / \         /X\ / \ / \
    o---o---o---o       o---o---o---o       o---o---o---o
   / \ / \ /X\ / \     / \ / \ / \X/ \     / \ / \ / \X/ \
  o---o---o---o---o   o---o---o---o---o   o---o---o---o---o
   \ / \ / \ / \X/     \ / \ / \ /X\ /     \ /X\ / \ / \ /
    o---o---o---o       o---o---o---o       o---o---o---o
     \X/ \ / \ /         \X/ \ / \ /         \ / \ / \X/
      o---o---o           o---o---o           o---o---o
   (12 symmetries)      (6 symmetries)      (6 symmetries)
For n = 2, the a(2) = 24 matrices counted are:
 1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
 2  3  4  1     2  3  4  1     2  4  1  3     2  4  3  1
 4  2  1  3     4  3  1  2     4  2  3  1     4  1  2  3
-
 1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
 2  4  3  1     3  1  4  2     3  2  4  1     3  2  4  1
 4  2  1  3     3  4  1  2     3  4  1  2     4  3  1  2
-
 1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
 3  4  1  2     3  4  1  2     3  4  2  1     3  4  2  1
 4  1  3  2     4  2  3  1     4  1  2  3     4  1  3  2
plus the same matrices with rows 2 and 3 interchanged.

Cf. A002047, A000170, A000142.

% minizinc -D 'N=5' -s --all-solutions a375800.mzn
include "globals.mzn";
include "alldifferent.mzn";
int: N;
array[1..2*N] of var 1..2*N: perm1;
array[1..2*N] of var 1..2*N: perm2;
constraint forall(i in 1..2*N)(3*N+1 <= perm1[i]+perm2[i]+i /\ perm1[i]+perm2[i]+i <= 3*N+2);
constraint alldifferent(perm1);
constraint alldifferent(perm2);
solve satisfy;
output [show(i) ++ " " | i in 1..2*N];
output [show(perm1[i]) ++ " " | i in 1..2*N];
output [show(perm2[i]) ++ " " | i in 1..2*N];

A377700 Number of ways of placing n nonattacking rooks on a toroidal board of 2n^2 equilateral triangular spaces.

Original entry on oeis.org

2, 4, 6, 48, 30, 1152, 266, 45824, 4050, 2736000, 75702, 233017344, 2060734

Offset: 1

Hugh Robinson, Nov 04 2024

The board is formed by identifying opposite edges of a 60°/120° rhombus, tiled by equilateral triangles with n triangles along each edge.
A rook move on the equilateral triangle tessellation is a move along a path through successively edge-adjacent faces, that turns alternately left and right at each face (starting with either left or right), so that the overall direction of movement remains approximately parallel to one set of triangle edges.
Also counts the number of 3 X n matrices such that each row is a permutation of {1, ..., n}, the first row is the identity permutation (1 .. n), and each column sums to either 0 or 1 (mod n).
When n = 2k+1, the rooks must be either all on white spaces or all on black spaces (using the obvious parity coloring) and the problem is equivalent to A006717(k) for each case. When n = 2k, there are k rooks on white spaces and k rooks on black spaces.

			For n = 4, the a(4) = 48 arrangements are generated from one solution by symmetries of the toroidal grid:
              o---o---o---o---o
             / \ /X\ / \ / \ /
            o---o---o---o---o
           / \X/ \ / \ / \ /
          o---o---o---o---o
         / \ / \ / \ / \X/
        o---o---o---o---o
       / \ / \ /X\ / \ /
      o---o---o---o---o

Cf. A375800, A006717, A002047, A062166, A067015, A342372.

% minizinc -D 'N=6' -s --all-solutions a.mzn
include "globals.mzn";
include "alldifferent.mzn";
int: N;
array[1..N] of var 1..N: perm1;
array[1..N] of var 1..N: perm2;
constraint alldifferent(perm1);
constraint alldifferent(perm2);
constraint forall(i in 1..N)(perm1[i] + perm2[i] + i in {N,N+1,2*N,2*N+1,3*N});
solve satisfy;
output [show(i) ++ " " | i in 1..N];
output [show(perm1[i]) ++ " " | i in 1..N];
output [show(perm2[i]) ++ " " | i in 1..N];

a(2k+1) = 2 * A006717(k).

cp's OEIS Frontend

A375800 Number of ways of placing 2n nonattacking rooks on a hexagonal board of equilateral triangular spaces with n spaces along each edge.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

MiniZinc