A178446 Number of perfect matchings in the n X n X n triangular grid, reduced by the spire vertex if n mod 4 equals 1 or 2.

1, 1, 1, 2, 6, 28, 200, 2196, 37004, 957304, 38016960, 2317631400, 216893681800, 31159166587056, 6871649018572800, 2326335506123418128, 1208982377794384163088, 964503557426086478029152, 1181201363574177619007442944, 2220650888749669503773432361504, 6408743336016148761893699822360672

Offset: 0

Author

Alois P. Heinz, Dec 24 2010

Keywords

• nonn

Comments

The n X n X n triangular grid has n rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(n) vertices and 3*A000217(n-1) edges altogether.

In order to be able to find matchings the n X n X n triangular grid is reduced by the spire vertex (one vertex in row 1) and the incident edges if n mod 4 is in {1, 2}. The resulting graph has an even number of vertices.

Examples

			4 example graphs:                         o
                                         / \
                             o          o---o
                            / \        / \ / \
                   ( )     o---o      o---o---o
                          / \ / \    / \ / \ / \
              ( ) o---o  o---o---o  o---o---o---o
  n:           1    2        3            4
  Vertices:    0    2        6           10
  Edges:       0    1        9           18
  Matchings:   1    1        2            6
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
• Eric Weisstein's World of Mathematics, Perfect Matching
• Wikipedia, Triangular grid graph

Crossrefs

Cf. A039907, A071093, A182797.

Programs

Maple

with(LinearAlgebra):
a:= proc(n) option remember; local i, j, h0, h1, M, s, t;
      if n<2 then 1
    else s:= `if`(member(irem(n, 4), [1, 2]), 1, 0);
         M:= Matrix((n+1)*n/2 -s, shape=skewsymmetric);
         if s=1 then M[1,2]:=1 fi;
         for j from 1+s to n-1 do
           h0:= j*(j-1)/2 +1-s;
           h1:= h0+j;
           t:= 1;
           for i from 1 to j do
             M[h1, h1+1]:= 1;
             M[h1, h0]:= t;
             h1:= h1+1;
             M[h1, h0]:= t;
             h0:= h0+1;
             t:= -t
           od
         od;
         sqrt(Determinant(M))
      fi
    end:
seq(a(n), n=0..15);

Mathematica

a[n_] := a[n] = Module[{i, j, h0, h1, M, s, t}, If[n<2, 1, s = If[1 <= Mod[n, 4] <= 2, 1, 0]; M = Array[0&, {(n+1)n/2 - s, (n+1)n/2 - s}]; If[s == 1, M[[1, 2]] = 1]; For[j = 1+s, j <= n-1, j++, h0 = j(j-1)/2 + 1 - s; h1 = h0+j; t = 1; For[i = 1, i <= j, i++, M[[h1, h1+1]] = 1; M[[h1, h0]] = t; h1 = h1+1; M[[h1, h0]] = t; h0 = h0+1; t = -t]]; Sqrt[Det[M-Transpose[M]]]]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 23 2022, after Alois P. Heinz *)