A182368 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square grid graph G_(n,n), highest powers first.

1, 0, 1, -4, 6, -3, 0, 1, -12, 66, -216, 459, -648, 594, -323, 79, 0, 1, -24, 276, -2015, 10437, -40614, 122662, -292883, 557782, -848056, 1022204, -960627, 682349, -346274, 112275, -17493, 0, 1, -40, 780, -9864, 90798, -647352, 3714180, -17590911, 69997383

Offset: 1

Author

Alois P. Heinz, Apr 26 2012

Keywords

• sign
• look
• tabf
• hard

Comments

The square grid graph G_(n,n) has n^2 = A000290(n) vertices and 2*n*(n-1) = A046092(n-1) edges. The chromatic polynomial of G_(n,n) has n^2+1 = A002522(n) coefficients.

Examples

			3 example graphs:                          o---o---o
.                                          |   |   |
.                             o---o        o---o---o
.                             |   |        |   |   |
.                o            o---o        o---o---o
Graph:        G_(1,1)        G_(2,2)        G_(3,3)
Vertices:        1              4              9
Edges:           0              4             12
The square grid graph G_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
  1,    0;
  1,   -4,     6,      -3,        0;
  1,  -12,    66,    -216,      459,       -648,         594, ...
  1,  -24,   276,   -2015,    10437,     -40614,      122662, ...
  1,  -40,   780,   -9864,    90798,    -647352,     3714180, ...
  1,  -60,  1770,  -34195,   486210,   -5421612,    49332660, ...
  1,  -84,  3486,  -95248,  1926585,  -30755376,   403410654, ...
  1, -112,  6216, -227871,  6205479, -133865298,  2382122274, ...
  1, -144, 10296, -487280, 17169852, -480376848, 11114098408, ...
  ...
		

Links

• Alois P. Heinz, Rows n = 1..9, flattened
• Eric Weisstein's World of Mathematics, Chromatic Polynomial
• Eric Weisstein's World of Mathematics, Grid Graph
• Wikipedia, Chromatic Polynomial

Crossrefs

Columns 0, 1 give: A000012, (-1)*A046092(n-1).

Sums of absolute values of row elements give: A080690(n).

Cf. A000290, A002522, A182406, A185442, A193233, A193277, A193283.

Programs

Mathematica

Reverse /@ CoefficientList[Table[ChromaticPolynomial[GridGraph[{n, n}], x], {n, 5}], x] // Flatten (* Eric W. Weisstein, May 01 2017 *)