A246427 - OEIS

%I #25 Mar 24 2020 05:03:26

%S 10,70,896,52367

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

%C 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.

%C This also gives the degree of each anticut in the metric polytope (see link below) for n points.

%H A. Deza, <a href="http://www.cas.mcmaster.ca/~deza/metric.html">Metric Polytopes and Metric Cones</a>

%H P. Dukes, <a href="http://www.math.uvic.ca/~dukes/facets-tri9.txt">Nearly complete count of isomorphism types for n = 9</a>

%H P. Dukes and R. M. Wilson, <a href="http://dx.doi.org/10.1016/j.ejc.2006.07.008">The cone condition and t-designs</a>, European J. Combin. 28 (2007), 1610-1625.

%H Peter J. Dukes, K. Garaschuk, <a href="https://arxiv.org/abs/1608.06017">On the cone of weighted graphs generated by triangles</a>, arXiv preprint arXiv:1608.06017 [math.CO], 2016.

%H K. Garaschuk, <a href="http://hdl.handle.net/1828/5665">Linear methods for rational triangle decompositions</a>, Ph.D. Dissertation, University of Victoria, 2014.

%e 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.

%o (Sage)

%o def A246427(n):

%o     T = Combinations(range(n),2)

%o     K = Combinations(range(n),3)

%o     W = matrix(ZZ,binomial(n,2),binomial(n,3),lambda i,j:Set(T[i]).issubset(Set(K[j])))

%o     C = Cone(W.transpose())

%o     return len(C.facet_normals())

%o [A246427(n) for n in range(5,8)]

%Y Cf. A053043, A235459.

%K nonn,more

%O 5,1

%A _Peter J. Dukes_, Aug 26 2014