OFFSET
0,4
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.
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
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Dec 24 2010
STATUS
approved