254 research outputs found
Inequalities for the h- and flag h-vectors of geometric lattices
We prove that the order complex of a geometric lattice has a convex ear
decomposition. As a consequence, if D(L) is the order complex of a rank (r+1)
geometric lattice L, then for all i \leq r/2 the h-vector of D(L) satisfies
h(i-1) \leq h(i) and h(i) \leq h(r-i).
We also obtain several inequalities for the flag h-vector of D(L) by
analyzing the weak Bruhat order of the symmetric group. As an application, we
obtain a zonotopal cd-analogue of the Dowling-Wilson characterization of
geometric lattices which minimize Whitney numbers of the second kind. In
addition, we are able to give a combinatorial flag h-vector proof of h(i-1)
\leq h(i) when i \leq (2/7)(r + 5/2).Comment: 15 pages, 2 figures. Typos fixed; most notably in Table 1. A note was
added regarding a solution to problem 4.
Projection volumes of hyperplane arrangements
We prove that for any finite real hyperplane arrangement the average
projection volumes of the maximal cones is given by the coefficients of the
characteristic polynomial of the arrangement. This settles the conjecture of
Drton and Klivans that this held for all finite real reflection arrangements.
The methods used are geometric and combinatorial. As a consequence we determine
that the angle sums of a zonotope are given by the characteristic polynomial of
the order dual of the intersection lattice of the arrangement
Enumerative properties of triangulations of spherical bundles over S^1
We give a complete characterization of all possible pairs (v,e), where v is
the number of vertices and e is the number of edges, of any simplicial
triangulation of an S^k-bundle over S^1. The main point is that Kuhnel's
triangulations of S^{2k+1} x S^1 and the nonorientable S^{2k}-bundle over S^1
are unique among all triangulations of (n-1)-dimensional homology manifolds
with first Betti number nonzero, vanishing second Betti number, and 2n+1
vertices.Comment: To appear in European J. of Combinatorics. Many typos fixe
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