A260032 Number of perfect matchings in graph P_{2n} X P_{2n} with a monomer on each corner.

1, 8, 784, 913952, 12119367744, 1773206059548800, 2808001509386950713600, 47534638766423741578738188800, 8530835766072904609739799813424153600, 16137081911409285302469685272022812457875802112, 320397648203287990193211938297925486964232264783587250176

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..40

N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014, p.21.

Wikipedia, FKT algorithm

Wikipedia, Matching (graph theory)

Maple

with(LinearAlgebra):
a:= proc(n) option remember; local d, i, j, t, m, M;
      d:= 2*n; m:= d^2-4;
      M:= Matrix(m, shape=skewsymmetric);
      for i to d-3 do M[i+1, i]:=1 od;
      for i to d-2 do M[i, i+d-1]:=1 od;
      for i from m-d+3 to m-1 do M[i, i+1]:=1 od;
      for i from m-d+3 to m do M[i-d+1, i]:=1 od;
      for i from d-1 to m-2*d+2 do M[i, i+d]:=1 od;
      for i to d-2 do for j to d-1 do
        t:=d*i+j-2; M[t, t+1]:= `if`(irem(i, 2)=1, 1, -1);
      od od;
      isqrt(Determinant(M))
    end:
seq(a(n), n=1..11);  # Alois P. Heinz, Mar 10 2016

Mathematica

a[1] = 1; a[n_] := a[n] = Module[{d, i, j, t, m, M}, d = 2*n; m = d^2 - 4; M = Array[0&, {m, m}];
   For[i = 1, i <= d - 3, i++, M[[i + 1, i]] = 1];
   For[i = 1, i <= d - 2, i++, M[[i, i + d - 1]] = 1];
   For[i = m - d + 3, i <= m - 1, i++, M[[i, i + 1]] = 1];
   For[i = m - d + 3, i <= m, i++, M[[i - d + 1, i]] = 1];
   For[i = d - 1, i <= m - 2*d + 2, i++, M[[i, i + d]] = 1];
   For[i = 1, i <= d - 2, i++,
    For[j = 1, j <= d - 1, j++, t = d*i + j - 2; M[[t, t + 1]] = If[Mod[i, 2] == 1, 1, -1]]]; M = M - Transpose[M]; Sqrt[Det[M]]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 11}] (* Jean-François Alcover, Nov 11 2017, after Alois P. Heinz *)

Crossrefs

Cf. A099390, A004003.

Sequence in context: A268148 A145415 A371595 * A332178 A204464 A001547

Adjacent sequences: A260029 A260030 A260031 * A260033 A260034 A260035

Keyword

nonn

Author

N. J. A. Sloane, Jul 19 2015

Extensions

a(6)-a(10) from Andrew Howroyd, Nov 15 2015

Typo in a(5) corrected and a(11) added by Alois P. Heinz, Mar 07 2016

Status

approved