Multiple planar coincidences with N-fold symmetry

The coincidence site lattices of the root lattice $A_4$ are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of $A_4$ into the icosian ring with its rich arithmetic structure, which recently (arXiv:math.MG/0702448) led to the classification of the similar sublattices of $A_4$.

The content of this preprint together with additional material appears now in 0706.2154.

This version has been withdrawn. The new and final version is on ArXiv 1103.4878

The lattice of noncrossing partitions is well-known for its wide variety of combinatorial appearances and properties. For example, the lattice is rank-symmetric and enumerated by the Catalan numbers. In this article, we introduce a large family of new noncrossing partition lattices with both of these properties, each parametrized by a configuration of n points in the plane.

Let $\J$ and $\K$ be convex sets in $\R^{n}$ whose affine spans intersect at a single rational point in $\J \cap \K$, and let $\J \oplus \K = \conv(\J \cup \K)$. We give formulas for the generating function {equation*} \sigma_{\cone(\J \oplus \K)}(z_1,..., z_n, z_{n+1}) = \sum_{(m_1,..., m_n) \in t(\J \oplus \K) \cap \Z^{n}} z_1^{m_1}... z_n^{m_n} z_{n+1}^{t} {equation*} of lattice points in all integer dilates of $\J \oplus \K$ in terms of $\sigma_{\cone \J}$ and $\sigma_{\c...

Lattice paths effectively model phenomena in chemistry, physics and probability theory. Asymptotic enumeration of lattice paths is linked with entropy in the physical systems being modeled. Lattice paths restricted to different regions of the plane are well suited to a functional equation approach for exact and asymptotic enumeration. This thesis surveys results on lattice paths under various restrictions, with an emphasis on lattice paths in the quarter plane. For these path...

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Cz\'edli and E.T. Schmidt as the duals of the lattices consisting of the intersections of the members of two composition series in a group. Our main result determines the number of (isomorphism classes of) these lattices of a given size in a recursive way. The corresponding planar diagrams, up to s...

The $GL_2$ Poincar\'{e} series giving the subconvexity results of Diaconu and Garrett is the solution to an automorphic partial differential equation, constructed by winding-up the solution to the corresponding differential equation on the free space. Generalizing this approach allows design of higher rank Poincar\'{e} series with specific number theoretic applications in mind: a Poincar\'{e} series for producing an explicit formula for the number of lattice points in an expa...

This doctoral thesis studies the overlap between two well-known collections of results in number theory: the theory of periods and period polynomials of modular forms as developed by Eichler, Shimura and Manin and its extensions by K\"ohnen and Zagier, and the theory of 'multiple zeta values' (MZV's) as initied by Euler and studied by many authors in the last two decades. These two theories had been linked by Manin, who introduced 'multiple L-values' (MLV's). We propose to st...

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a thir...