cp's OEIS Frontend

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

%I A246427 #25 Mar 24 2020 05:03:26
%S A246427 10,70,896,52367
%N A246427 Number of facets of the cone defined by the zero-one inclusion matrix of pairs versus triples on an n-set.
%C A246427 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 A246427 This also gives the degree of each anticut in the metric polytope (see link below) for n points.
%H A246427 A. Deza, <a href="http://www.cas.mcmaster.ca/~deza/metric.html">Metric Polytopes and Metric Cones</a>
%H A246427 P. Dukes, <a href="http://www.math.uvic.ca/~dukes/facets-tri9.txt">Nearly complete count of isomorphism types for n = 9</a>
%H A246427 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 A246427 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 A246427 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 A246427 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 A246427 (Sage)
%o A246427 def A246427(n):
%o A246427     T = Combinations(range(n),2)
%o A246427     K = Combinations(range(n),3)
%o A246427     W = matrix(ZZ,binomial(n,2),binomial(n,3),lambda i,j:Set(T[i]).issubset(Set(K[j])))
%o A246427     C = Cone(W.transpose())
%o A246427     return len(C.facet_normals())
%o A246427 [A246427(n) for n in range(5,8)]
%Y A246427 Cf. A053043, A235459.
%K A246427 nonn,more
%O A246427 5,1
%A A246427 _Peter J. Dukes_, Aug 26 2014