cp's OEIS Frontend

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

%I A377700 #11 Nov 13 2024 08:29:40
%S A377700 2,4,6,48,30,1152,266,45824,4050,2736000,75702,233017344,2060734
%N A377700 Number of ways of placing n nonattacking rooks on a toroidal board of 2n^2 equilateral triangular spaces.
%C A377700 The board is formed by identifying opposite edges of a 60°/120° rhombus, tiled by equilateral triangles with n triangles along each edge.
%C A377700 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.
%C A377700 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).
%C A377700 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.
%F A377700 a(2k+1) = 2 * A006717(k).
%e A377700 For n = 4, the a(4) = 48 arrangements are generated from one solution by symmetries of the toroidal grid:
%e A377700               o---o---o---o---o
%e A377700              / \ /X\ / \ / \ /
%e A377700             o---o---o---o---o
%e A377700            / \X/ \ / \ / \ /
%e A377700           o---o---o---o---o
%e A377700          / \ / \ / \ / \X/
%e A377700         o---o---o---o---o
%e A377700        / \ / \ /X\ / \ /
%e A377700       o---o---o---o---o
%o A377700 (MiniZinc)
%o A377700 % minizinc -D 'N=6' -s --all-solutions a.mzn
%o A377700 include "globals.mzn";
%o A377700 include "alldifferent.mzn";
%o A377700 int: N;
%o A377700 array[1..N] of var 1..N: perm1;
%o A377700 array[1..N] of var 1..N: perm2;
%o A377700 constraint alldifferent(perm1);
%o A377700 constraint alldifferent(perm2);
%o A377700 constraint forall(i in 1..N)(perm1[i] + perm2[i] + i in {N,N+1,2*N,2*N+1,3*N});
%o A377700 solve satisfy;
%o A377700 output [show(i) ++ " " | i in 1..N];
%o A377700 output [show(perm1[i]) ++ " " | i in 1..N];
%o A377700 output [show(perm2[i]) ++ " " | i in 1..N];
%Y A377700 Cf. A375800, A006717, A002047, A062166, A067015, A342372.
%K A377700 nonn,more
%O A377700 1,1
%A A377700 _Hugh Robinson_, Nov 04 2024