A185442 Triangle T(n,k), n>=1, 0<=k<=2n(n+1), read by rows: row n gives the coefficients of the chromatic polynomial of the Aztec diamond graph of order n, highest powers first.

1, -4, 6, -3, 0, 1, -16, 120, -555, 1755, -3978, 6588, -7965, 6885, -4050, 1458, -243, 0, 1, -36, 630, -7127, 58476, -370128, 1876942, -7818056, 27208798, -80059990, 200769740, -431267475, 795531116, -1260437072, 1711682175, -1983112401, 1945239399, -1597006926, 1079055243, -585362106, 245489859, -74816136, 14762007, -1416933, 0

Offset: 1

Author

Alois P. Heinz, Feb 03 2011

Keywords

• sign
• tabf
• look
• hard

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.

Examples

			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,  ...
  ...
		

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

Crossrefs

Cf. A046092, A016742, A006125, A007725.