A178435 Number of acyclic orientations of the n X n X n triangular grid.

1, 6, 162, 19602, 10619910, 25753129470, 279488630719746, 13573527285845525634, 2949851294016821586137934, 2868652614504623418332698354038, 12483073717920041560887416137620435882, 243068197882943244196175524589364487906969746, 21178547618859581967063811182618272071362317831449326

Offset

1,2

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.

An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.

Links

Table of n, a(n) for n=1..13.

Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8

Wikipedia, Chromatic polynomial

Wikipedia, Triangular grid graph

Crossrefs

Cf. A182797, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796, A182798, A000217.

Sequence in context: A362732 A078535 A177781 * A183254 A143534 A104729

Adjacent sequences: A178432 A178433 A178434 * A178436 A178437 A178438

Keyword

hard,nonn

Author

Alois P. Heinz, Dec 21 2010

Status

approved