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A189006 Array A(m,n) read by antidiagonals: number of domino tilings of the m X n grid with upper left corner removed iff m*n is odd, (m>=0, n>=0). 8
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 5, 1, 1, 1, 1, 8, 11, 11, 8, 1, 1, 1, 1, 13, 15, 36, 15, 13, 1, 1, 1, 1, 21, 41, 95, 95, 41, 21, 1, 1, 1, 1, 34, 56, 281, 192, 281, 56, 34, 1, 1, 1, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 1, 1, 1, 89, 209, 2245, 2415, 6728, 2415, 2245, 209, 89, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

LINKS

Alois P. Heinz, Antidiagonals n = 0..75, flattened

Eric Weisstein's World of Mathematics, Perfect Matching

Wikipedia, FKT algorithm

Wikipedia, Matching (graph theory)

EXAMPLE

A(3,3) = 4, because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:

  . .___. . .___. . .___. . .___.

  ._|___| ._|___| ._| | | ._|___|

  | |___| | | | | | |_|_| |___| |

  |_|___| |_|_|_| |_|___| |___|_|

Array begins:

  1, 1,  1,  1,   1,    1,    1, ...

  1, 1,  1,  1,   1,    1,    1, ...

  1, 1,  2,  3,   5,    8,   13, ...

  1, 1,  3,  4,  11,   15,   41, ...

  1, 1,  5, 11,  36,   95,  281, ...

  1, 1,  8, 15,  95,  192, 1183, ...

  1, 1, 13, 41, 281, 1183, 6728, ...

MAPLE

with(LinearAlgebra):

A:= proc(m, n) option remember; local i, j, s, t, M;

      if m=0 or n=0 then 1

    elif m<n then A(n, m)

    else s:= irem(n*m, 2);

         M:= Matrix(n*m-s, shape=skewsymmetric);

         for i to n do

           for j to m do

             t:= (i-1)*m+j-s;

             if i>1 or j>1 or s=0 then

               if j<m then M[t, t+1]:= 1 fi;

               if i<n then M[t, t+m]:= 1-2*irem(j, 2) fi

             fi

           od

         od;

         isqrt(Determinant(M))

      fi

    end:

seq(seq(A(m, d-m), m=0..d), d=0..15);

MATHEMATICA

A[1, 1] = 1; A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2]; M[i_, j_] /; j < i := -M[j, i]; M[_, _] = 0; For[i = 1, i <= n, i++, For[j = 1, j <= m, j++, t = (i-1)*m+j-s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t+1] = 1]; If[i < n, M[t, t+m] = 1-2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m-s, n*m-s}]]]]]; Table[Table[A[m, d-m], {m, 0, d}], {d, 0, 15}] // Flatten (* Jean-Fran├žois Alcover, Dec 26 2013, translated from Maple *)

CROSSREFS

Rows m=0+1, 2-12 give: A000012, A000045(n+1), A002530(n+1), A005178(n+1), A189003, A028468, A189004, A028470, A189005, A028472, A210724, A028474.

Main diagonal gives: A189002.

Cf. A099390, A187596, A187616, A187617, A187618, A004003.

Sequence in context: A193517 A296554 A327482 * A245013 A219924 A226444

Adjacent sequences:  A189003 A189004 A189005 * A189007 A189008 A189009

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Apr 15 2011

STATUS

approved

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Last modified October 18 05:45 EDT 2019. Contains 328146 sequences. (Running on oeis4.)