OFFSET
5,1
COMMENTS
Equivalently, this is the number of integer weightings of the edges of the complete graph K_n which are: (1) nonnegative on all triangles; (2) maximally vanishing on triangles; and (3) have gcd of weights equal to one.
This also gives the degree of each anticut in the metric polytope (see link below) for n points.
LINKS
A. Deza, Metric Polytopes and Metric Cones
P. Dukes and R. M. Wilson, The cone condition and t-designs, European J. Combin. 28 (2007), 1610-1625.
Peter J. Dukes, K. Garaschuk, On the cone of weighted graphs generated by triangles, arXiv preprint arXiv:1608.06017 [math.CO], 2016.
K. Garaschuk, Linear methods for rational triangle decompositions, Ph.D. Dissertation, University of Victoria, 2014.
EXAMPLE
For n = 5, the 10 facet normals are defined by the choice of a (2,3)-partition. Weight 2 is assigned to edges within each part and weight -1 is assigned to edges crossing the partition. Every triangle has weight 0, except for one which inherits weight 6.
PROG
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Peter J. Dukes, Aug 26 2014
STATUS
approved