A235459 -id:A235459 - cp's OEIS frontend

A053043 Number of facets of hypermetric correlation cone.

3, 12, 40, 210, 3773

Offset: 2

N. J. A. Sloane, Feb 25 2000

In Table 1 of Deza & Dutour Sikirić 2018 (also in Table 13.1 on p. 223 of "Generalizations of Finite Metrics and Cuts"), the same initial terms are given for the number of facets of the hypermetric cone HYP_{n-1}, with the next term 298592. However, the hypermetric cone HYP and the hypermetric correlation cone HYCO apparently have different definitions. - Andrey Zabolotskiy, Jun 25 2022

M. Deza and V. Grishukhin, Voronoi L-decomposition of PSD_n and the hypermetric correlation cone, pp. 75-79 of Voronoi's Impact on Modern Science, ed. P. Engel and H. Syta, 2 vols, Inst. Math., Nat. Acad. Sci. Ukraine, Kiev 1998.
Elena Deza, Michel Deza and Mathieu Dutour Sikirić, Generalizations of Finite Metrics and Cuts, World Scientific, 2016.

Michel Deza and Mathieu Dutour Sikirić, The hypermetric cone and polytope on eight vertices and some generalizations, Journal of Symbolic Computation, 88 (2018), 67-84.

Cf. A235459.

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

Original entry on oeis.org

10, 70, 896, 52367

Offset: 5

Peter J. Dukes, Aug 26 2014

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.

			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.

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.

Cf. A053043, A235459.

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)]

cp's OEIS Frontend

A053043 Number of facets of hypermetric correlation cone.

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