A246427 Number of facets of the cone defined by the zero-one inclusion matrix of pairs versus triples on an n-set.

10, 70, 896, 52367

Offset: 5

Author

Peter J. Dukes, Aug 26 2014

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

• A. Deza, Metric Polytopes and Metric Cones
• P. Dukes, Nearly complete count of isomorphism types for n = 9
• 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.

Crossrefs

Cf. A053043, A235459.

Programs

Sage

def A246427(n):
    T = Combinations(range(n),2)
    K = Combinations(range(n),3)
    W = matrix(ZZ,binomial(n,2),binomial(n,3),lambda i,j:Set(T[i]).issubset(Set(K[j])))
    C = Cone(W.transpose())
    return len(C.facet_normals())
[A246427(n) for n in range(5,8)]