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

%I A375800 #17 Oct 21 2024 14:35:19
%S A375800 3,24,348,7812,255756,11747504,714121392
%N A375800 Number of ways of placing 2n nonattacking rooks on a hexagonal board of equilateral triangular spaces with n spaces along each edge.
%C A375800 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 A375800 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.
%e A375800 For n = 2, the a(2) = 24 arrangements are rotations and reflections of:
%e A375800       o---o---o           o---o---o           o---o---o
%e A375800      /X\ / \ / \         /X\ / \ / \         /X\ / \ / \
%e A375800     o---o---o---o       o---o---o---o       o---o---o---o
%e A375800    / \ / \ /X\ / \     / \ / \ / \X/ \     / \ / \ / \X/ \
%e A375800   o---o---o---o---o   o---o---o---o---o   o---o---o---o---o
%e A375800    \ / \ / \ / \X/     \ / \ / \ /X\ /     \ /X\ / \ / \ /
%e A375800     o---o---o---o       o---o---o---o       o---o---o---o
%e A375800      \X/ \ / \ /         \X/ \ / \ /         \ / \ / \X/
%e A375800       o---o---o           o---o---o           o---o---o
%e A375800    (12 symmetries)      (6 symmetries)      (6 symmetries)
%e A375800 For n = 2, the a(2) = 24 matrices counted are:
%e A375800  1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
%e A375800  2  3  4  1     2  3  4  1     2  4  1  3     2  4  3  1
%e A375800  4  2  1  3     4  3  1  2     4  2  3  1     4  1  2  3
%e A375800 -
%e A375800  1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
%e A375800  2  4  3  1     3  1  4  2     3  2  4  1     3  2  4  1
%e A375800  4  2  1  3     3  4  1  2     3  4  1  2     4  3  1  2
%e A375800 -
%e A375800  1  2  3  4     1  2  3  4     1  2  3  4     1  2  3  4
%e A375800  3  4  1  2     3  4  1  2     3  4  2  1     3  4  2  1
%e A375800  4  1  3  2     4  2  3  1     4  1  2  3     4  1  3  2
%e A375800 plus the same matrices with rows 2 and 3 interchanged.
%o A375800 (MiniZinc)
%o A375800 % minizinc -D 'N=5' -s --all-solutions a375800.mzn
%o A375800 include "globals.mzn";
%o A375800 include "alldifferent.mzn";
%o A375800 int: N;
%o A375800 array[1..2*N] of var 1..2*N: perm1;
%o A375800 array[1..2*N] of var 1..2*N: perm2;
%o A375800 constraint forall(i in 1..2*N)(3*N+1 <= perm1[i]+perm2[i]+i /\ perm1[i]+perm2[i]+i <= 3*N+2);
%o A375800 constraint alldifferent(perm1);
%o A375800 constraint alldifferent(perm2);
%o A375800 solve satisfy;
%o A375800 output [show(i) ++ " " | i in 1..2*N];
%o A375800 output [show(perm1[i]) ++ " " | i in 1..2*N];
%o A375800 output [show(perm2[i]) ++ " " | i in 1..2*N];
%Y A375800 Cf. A002047, A000170, A000142.
%K A375800 nonn,more
%O A375800 1,1
%A A375800 _Hugh Robinson_, Aug 29 2024