A Möbius Identity Arising from Modularity in a Matroid Bilinear Form

The matrix for the bilinear form of the flag space of a matroid has (with respect to an appropriate basis) a tensor product structure when the matroid has a modular flat . When determinants are taken, an identity is obtained for the rho function (a certain product of the Möbius and beta functions) summed over flats with a fixed intersection with . When the identity is interpreted for Dowling lattices and finite projective spaces, identities with similar combinatorial proofs are obtained for binomial and Gaussian coefficients, respectively

Generalized Integer Partitions, Tilings of Zonotopes and Lattices

In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of zonotopes, ordered with a simple and classical dynamics, is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical systems exist. These results give a better understanding of the behaviour of tilings of zonotopes with flips and dynamical systems involving partitions.Comment: See http://www.liafa.jussieu.fr/~latapy

The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems

We show that an effective version of Siegel’s Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the κ ≥ 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. This dichotomy holds even when restricted to planar graphs. A special case of this result is that counting edge κ-colorings is #P-hard over planar 3-regular graphs for κ ≥ 3. In fact, we prove that counting edge κ-colorings is #P-hard over planar r-regular graphs for all κ ≥ r ≥ 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.

Contents

2 Preliminary de nitions and notation 3 3 Schedulablity and a bound on utilization

Some applications of the intersection theory to Galois geometry.

The authors state several identities and inequalities for the intersection matrix IS of a matroid S embedded in a projective space PG(n,q) . These conditions are used to prove results due to Bruck, Bose, the reviewer and Qvist on the sizes of subplanes, arcs and caps embedded in a projective space PG(n,q) . The paper is very clearly written and its results provide new methods for solving combinatorial problems of Galois geometries