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

5,1

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

Table of n, a(n) for n=5..8.

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.

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

(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)]

Crossrefs

Cf. A053043, A235459.

Sequence in context: A037705 A126505 A038779 * A172499 A215295 A136856

Adjacent sequences: A246424 A246425 A246426 * A246428 A246429 A246430

Keyword

nonn,more

Author

Peter J. Dukes, Aug 26 2014

Status

approved