OFFSET
1,2
COMMENTS
The Aztec diamond graph of order n has 2*n*(n+1) vertices with integer coordinates (x,y) obeying |x-1/2| + |y-1/2| <= n and (2*n)^2 edges connecting vertices having Euclidean distance 1. It can be derived from the Aztec diamond using vertices to represent tiles and edges to connect vertices of neighboring tiles. The chromatic polynomial has 2*n*(n+1)+1 coefficients.
LINKS
Alois P. Heinz, Rows n = 1..7, flattened
Propp, J., Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
Eric Weisstein's World of Mathematics, Aztec Diamond
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Wikipedia, Chromatic Polynomial
EXAMPLE
2 example graphs: o-o
. | |
. o-o-o-o
. | | | |
. o-o o-o-o-o
. | | | |
. o-o o-o
Order: 1 2
Vertices: 4 12
Edges: 4 16
The Aztec diamond graph of order 1 is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
1, -4, 6, -3, 0;
1, -16, 120, -555, 1755, -3978, 6588, ...
1, -36, 630, -7127, 58476, -370128, 1876942, ...
1, -64, 2016, -41639, 633851, -7578762, 74074918, ...
1, -100, 4950, -161659, 3917248, -75096624, 1186008180, ...
1, -144, 10296, -487283, 17170275, -480406458, 11115470152, ...
CROSSREFS
KEYWORD
AUTHOR
Alois P. Heinz, Feb 03 2011
STATUS
approved