Enumeration of tilings of diamonds and hexagons with defects

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings

On growth in an abstract plane

There is a parallelism between growth in arithmetic combinatorics and growth in a geometric context. While, over $\mathbb{R}$ or $\mathbb{C}$, geometric statements on growth often have geometric proofs, what little is known over finite fields rests on arithmetic proofs. We discuss strategies for geometric proofs of growth over finite fields, and show that growth can be defined and proven in an abstract projective plane -- even one with weak axioms

Explicit L2L^2 bounds for the Riemann ζ\zeta function

Bounds on the tails of the zeta function $\zeta$, and in particular explicit bounds, are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. An explicit version of the traditional `convexity bound' has long been known (Backlund 1918). To do better, one must either provide explicit versions of subconvexity bounds, or give explicit bounds on means of $\zeta$. Here we take the second road, bounding weighted $L^2$ norms of tails of $\zeta$. Two approaches are followed, each giving the better result on a different range. One of them is inspired by the proof of the standard mean value theorem for Dirichlet polynomials (Montgomery 1971). The main technical idea is the use of a carefully chosen smooth approximation to $1_{\lbrack 0,1\rbrack}$ so as to eliminate off-diagonal terms. The second approach, superior for large $T$, is based on classical lines, starting with an approximation to $\zeta$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<\sigma\leq 1$ and are strong enough to be of practical use in giving precise values for integrals when combined with (rigorous) numerical integration. We also present bounds for the $L^{2}$ norm of $\zeta$ in $[1,T]$ for $0\leq\sigma\leq 1$.Comment: 44 pages; v6: corrections, improvements and clarifications, added reference

Bounds on the diameter of Cayley graphs of the symmetric group

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.Comment: 17 pages, 6 table

Growth in solvable subgroups of GL_r(Z/pZ)

Let $K=Z/pZ$ and let $A$ be a subset of \GL_r(K) such that is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifically we prove that either $A$ grows rapidly (meaning $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), or else there are groups $U_R$ and $S$, with $S/U_R$ nilpotent such that $A_k\cap S$ is large and $U_R\subseteq A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k : x_i \in A \cup A^{-1} \cup {1}}$. The implied constants depend only on the rank $r$ of $\GL_r(K)$. When combined with recent work by Pyber and Szab\'o, the main result of this paper implies that it is possible to draw the same conclusions without supposing that is solvable.Comment: 46 pages. This version includes revisions recommended by an anonymous referee including, in particular, the statement of a new theorem, Theorem