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Revision History for A212194

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A212194

Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the staggered hexagonal square grid graph SH_(n,n), highest powers first.
(history; published version)

#14 by Alois P. Heinz at Wed Feb 08 18:31:03 EST 2017

STATUS

editing

approved

#13 by Alois P. Heinz at Wed Feb 08 18:30:59 EST 2017

LINKS

Wikipedia, <a href="httphttps://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>

STATUS

approved

editing

#12 by N. J. A. Sloane at Sat May 05 17:33:22 EDT 2012

STATUS

proposed

approved

#11 by Alois P. Heinz at Sat May 05 15:16:37 EDT 2012

STATUS

editing

proposed

#10 by Alois P. Heinz at Fri May 04 17:47:30 EDT 2012

LINKS

Wikipedia, <a href="http://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>

#9 by Alois P. Heinz at Thu May 03 20:28:01 EDT 2012

EXAMPLE

1, -56, 1508, -25992, 321994, -3051871, ... , -3101089710, ...

#8 by Alois P. Heinz at Thu May 03 20:18:48 EDT 2012

COMMENTS

The staggered hexagonal square grid graph SH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges. The chromatic polynomial of SH_(n,n) has n^2+1 = A002522(n) coefficients.

EXAMPLE

The staggered hexagonal square grid graph SH_(2,2) has chromatic polynomial q^4 -5*q^3 +8*q^2 -4*q => row 2 = [1, -5, 8, -4, 0].

Triangle T(n,k) begins:

1, 0;

1, -5, 8, -4, 0;

1, -16, 112, -448, 1120, -1791, ...

1, -33, 510, -4898, 32703, -160859, ...

1, -56, 1508, -25992, 321994, -3051871, ...

1, -85, 3520, -94620, 1855860, -28306676, ...

1, -120, 7068, -272344, 7720110, -171656543, ...

1, -161, 12782, -667058, 25738055, -783003395, ...

CROSSREFS

Columns 1-2 give: A000012, (-1)*A045944(n-1).

Row sums (for n>1) and last elements of rows give: A000004, row lengths give: A002522.

Cf. A000290, A212162, A212195.

#7 by Alois P. Heinz at Thu May 03 17:30:07 EDT 2012

COMMENTS

The staggered hexagonal square grid graph SH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges. The chromatic polynomial of SH_(n,n) has n^2+1 = A002522(n) coefficients.

#6 by Alois P. Heinz at Thu May 03 17:03:52 EDT 2012

EXAMPLE

3 example graphs: o--o--o

. | /|\ |

. |/ | \|

. o--o o--o--o

. | /| | /|\ |

. |/ | |/ | \|

. o o--o o--o--o

Graph: SH_(1,1) SH_(2,2) SH_(3,3)

Vertices: 1 4 9

Edges: 0 5 16

#5 by Alois P. Heinz at Thu May 03 16:38:42 EDT 2012

DATA

1, 0, 1, -5, 8, -4, 0, 1, -16, 112, -448, 1120, -1791, 1786, -1012, 248, 0, 1, -33, 510, -4898, 32703, -160859, 602408, -1749715, 3975561, -7068408, 9755858, -10265148, 7968348, -4304712, 1445104, -226720, 0, 1, -56, 1508, -25992, 321994, -3051871, 23000726, -141421592, 722137763, -3101089710

KEYWORD

sign,tabf,changed

Discussion

Thu May 03

16:39

Alois P. Heinz: A few terms more ... to show the difference.