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User: Hugh Robinson

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Hugh Robinson has authored 4 sequences.

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

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Author

Hugh Robinson, Nov 04 2024

Keywords

Comments

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.

Examples

			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

Crossrefs

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

Programs

  • MiniZinc
    % 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];

Formula

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

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

Original entry on oeis.org

3, 24, 348, 7812, 255756, 11747504, 714121392
Offset: 1

Views

Author

Hugh Robinson, Aug 29 2024

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A002047, A000170, A000142.

Programs

  • MiniZinc
    % 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];

A091378 Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the poset of order-preserving maps from [m] to [n+1] (where [m] denotes the total order on m objects), viewed as a category.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 96, 14, 1, 1, 42, 6560, 6560, 42, 1, 1, 132, 1738535, 771496766, 1738535, 132, 1, 1, 429, 2347585784
Offset: 0

Views

Author

Hugh Robinson, Mar 01 2004

Keywords

Comments

Specifying a weak factorization system on a poset category is equivalent to specifying a set of morphisms that includes all identity morphisms and is closed under composition and pullback.

Examples

			T(1, 2) = 5:  the category is the total order on three objects:  it has three nonidentity morphisms a, b, c satisfying the relation ba = c.  Of the 8 possible sets of morphisms, {a, b} is not closed under composition and {c}, {b, c} are not closed under pullback since a is a pullback of c.  The other 5 sets generate weak factorization systems.
See A092450 for an example computing weak factorization systems on a category which is not a total order.

Links

Crossrefs

Cf. A092450, A000108.

Formula

T(m, n) = T(n, m) because the corresponding categories are isomorphic. T(0, n) = T(n, 0) = 1. T(1, n) = T(n, 1) = C(n+1) the (n+1)st Catalan number (A000108).

Extensions

Corrected definition and more terms from Hugh Robinson, Oct 02 2011

A092450 Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the product category [m]x[n], where [m] denotes the total order on m objects, viewed as a category.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 10, 5, 1, 1, 14, 68, 68, 14, 1, 1, 42, 544, 1396, 544, 42, 1, 1, 132, 4828, 37434, 37434, 4828, 132, 1, 1, 429, 46124, 1226228, 4073836, 1226228, 46124, 429, 1, 1, 1430, 465932, 47002628, 645463414, 645463414, 47002628
Offset: 0

Views

Author

Hugh Robinson, Mar 24 2004

Keywords

Comments

Specifying a weak factorization system on a poset category is equivalent to specifying a set of morphisms that includes all identity morphisms and is closed under composition and pullback.

Examples

			T(2, 2) = 10:  the category has five nonidentity morphisms with relations ca = db = e.  a is a pullback of d and of e;  b is a pullback of c and of e.  So there are ten allowable sets of morphisms:  omitting identities for brevity, they are {}, {a}, {b}, {a,b}, {b,c}, {a,d}, {a,b,e}, {a,b,c,e}, {a,b,d,e}, {a,b,c,d,e}.

Links

Crossrefs

Cf. A091378, A000108.

Formula

T(0, n) = T(n, 0) = 1. T(1, n) = T(n, 1) = C(n) the n-th Catalan number (A000108).

Extensions

More terms from Hugh Robinson, Oct 02 2011

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