**George Andrews**, Penn State University
Title: False Theta Functions, Mock Theta Functions, and Orthogonal Polynomials

Recently, I have been looking at Bailey's Lemma and Bailey pairs from the point of view of orthogonal polynomials. This perspective has yielded a number of new combinatorial results mostly involving the Ramanujan's mock theta functions and the false theta functions of L.J.Rogers. We shall discuss the efficacy of this approach and shall apply it to various partition theoretic and combinatorial questions including "concave compositions" and "concatenatable spiral self-avoiding walks."

**Arvind Ayyer**, Institut de Physique Theorique, France
Title: Pattern avoidance and alternating sign matrices

We introduce a new class of objects, which we call _gog words_, that are in natural bijection with alternating sign matrices. We then define a notion of pattern avoidance on these words and moreover, show that gog words which avoid the pattern 312 are in natural bijection with a subset of totally symmetric self-complementary plane partitions.

Joint work with: Robert Cori

**Art Benjamin**, Harvey Mudd College
Title: The Combinatorialization of Linear Recurrences

We provide an original combinatorial proof of Binet's formula for
Fibonacci numbers
$$ F_n = (a^n - b^n)/\sqrt{5} $$
where $a = (1 + \sqrt{5})/2$ and $b = (1 - \sqrt{5})/2$.
Naturally, any k-th order linear recurrence with constant
coefficients has a
closed form solution, obtainable by factoring its (k-th degree)
characteristic polynomial. We extend our proof of Binet's formula to
show that these closed form solutions can also be given a combinatorial
interpretation, even in the repeated roots situtation. Based on joint
work with Halcyon Derks and Jennifer Quinn.

**Robert Brignall**, University of Bristol, UK
Title: Modular Decomposition and Graph Reconstruction

The Reconstruction Conjecture, one of the most well-known open problems in
Combinatorics, dates back to Kelly in 1957. It states that every graph on
at least 3 vertices can be determined uniquely by its multiset of
vertex-deleted subgraphs.

An easy argument using the concept of "attributability" proves that
disconnected graphs (and graphs whose complements are disconnected) can be
reconstructed. One natural generalisation of connectedness is to consider
decomposing a graph into its k-connected components, but the
attributability argument fails here when trying to "glue" these components
back together.
However, there are other ways to decompose a graph, and one which has
proved particularly useful in a wide range of contexts is the "modular
decomposition", which uniquely describes a graph in terms of smaller
"indecomposable" (or prime) graphs. In this talk, I will describe how the
attributability argument can be adapted to prove that many graphs which
have non-trivial modular decompositions (i.e. are not prime) can be
reconstructed.

Joint work with: Nic Georgiou and Rob Waters.

** Alexander Burstein**, Howard University
Title: On joint distribution of adjacencies, descents and some Mahonian
statistics

We consider the joint distribution on permutations of the number
of adjacencies (descents with consecutive values in consecutive
positions), descents and some Mahonian statistics. This refines, via a
direct bijection, some earlier results of Foata and Zeilberger (2001) with
a computer-aided proof.

** Sergi Elizalde**, Dartmouth College
Title: Descent sets of cyclic permutations

The descent set of a sequence $a_1a_2\dots$ is the set of indices $i$ such
that $a_i>a_{i+1}$. Consider the $n!$ cyclic permutations of
$\{1,2,\dots,n+1\}$ written in one-line notation, and for each one of them
remove the last entry $\pi(n+1)$. I will prove bijectively that the
descent sets of these objects have the same distribution as the descent
sets of permutations of $\{1,2,\dots,n\}$.

I will also discuss the problem that originated this work, which was the
study of permutations that can be obtained by lexicographically ordering
the successive shifts of an infinite word over an $N$-letter alphabet.

** Dominique Foata**, University of Strasbourg, France
Title:Tangent and Secant q-Calculus and (t,q)-Calculus

The connection between tangent numbers and Eulerian polynomials goes back to Euler (1754). The first combinatorial interpretations of those numbers are due to Andre' (1881) and of those polynomials to Riordan (1958). This connection can be carried over to a q-environment, even to a (t,q)-environment, both analytically and combinatorially.

** Aviezri Fraenkel**, Weizmann Institute of Science, Israel
Title: Ask Not What Doron Zeilberger Can Do For You, Ask What You Can Do For Doron Zeilberger

Two conjectures on multi-pile Wythoff's game that I once mentioned to Doron Zeilberger, were proved by him and Xinyu Sun very elegantly for the special case of 3 piles where the smallest pile has size at most 10, demonstrating that if you do something for Doron, you get a 10-fold reward! Their approach has the potential for an infinity-fold reward. We review the conjectures, the fractal-like structure of some complementary sets of integers connected with them (joint with Dalia Krieger), and new results and problems on complementary sets (in part joint with Peter Hegarty and Urban Larsson).

** Stavros Garoufalidis**, Georgia Tech
Title: Zeilberger meets Jones and Thurston

Eight years ago I had the pleasure to learn about q-holonomic sequences from Doron Zeilberger. It became immediate that the sequences of Jones polynomials of a knot and its parallels is q-holonomic. More generally, any natural sequence of a knotted 3-dimensional object in Quantum Topology is q-holonomic. I will talk about some basic geometric invariants of q-holonomic sequences, such as the characteristic variety (a plane curve in C^2, related to the specialization q=1), and the tropical curve (a tropical plane curve, related to the specialization q=0,infinity).

** Emilie Hogan**, Rutgers University
Title: Experimental techniques applied to convergence of rational difference equations

One of the questions when investigating rational difference equations, e.g.-
\[
x_{n+1} = \frac{ a_0+a_1 x_{n}+a_2 x_{n-1}+\cdots+a_{k+1}x_{n-k} }{ b_0+b_1 x_{n}+b_2 x_{n-1}+\cdots+b_{l+1}x_{n-l} }, \text{where} a_i, b_i \geq 0
\]
is whether or not they converge to an equilibrium given arbitrary positive initial conditions. There are some sufficient conditions for convergence of rational difference equations, but proving that a rational difference equation satisfies these sufficient conditions is often a daunting task itself. In this talk I will explain an experimental approach to showing convergence which involves verifying that repeated compositon of a function from $\mathbb{R}^{k+1}$ to $\mathbb{R}^{k+1}$ converges to the vector $\langle 0,\ldots,0\rangle \in \mathbb{R}^{k+1}$.

Joint work with Doron Zeilberger.

** Dennis Hou**, Rutgers University
Title: Extensions of Rowland's Prime-Generating Sequence

Eric Rowland has shown for suitable (and possibly all) n that
the sequence a(n)=a(n-1)+gcd(n,a(n-1)) in some sense naturally generates
primes, and it appears to belong to a broader class of such recurrences.
After surveying the variations of this sequence discovered by Benoit
Cloitre and Vladimir Shevelev, we discuss some further generalizations of
our own.

** Toufik Mansour**, University of Haifa, Israel
Title: Some recursive formulas for Selberg-type integrals

A set of recursive relations satisfied by Selberg-type integrals involving monomial symmetric polynomials are derived, generalizing well known results. These formulas provide a well-defined algorithm for computing Selberg-Schur
integrals whenever the Kostka numbers relating Schur functions and the corresponding monomial polynomials are explicitly known. We illustrate the usefulness of our results discussing some interesting examples.

Joint work with: Sergio Iguri (Instituto de Astronomia y Fisica del Espacio (Argentina).

** Luis Medina**, University of Puerto Rico
Title: Experiments with Exponential Sums over the Binary Field.

Let $\mathbb{F}$ be the binary field and $F({\bf X}) = F(X_1, \cdots, X_n)$ a polynomial in $n$ variables over $\mathbb{F}$. The exponential sum associated to $F$ over $\mathbb{F}$ is defined as
$$
S(F)=\sum_{x_1,\cdots,x_n \in \mathbb{F}}(-1)^{F(x_1,\cdots, x_n)}.
$$
Boolean functions (functions over $\mathbb{F}$) have many applications to cryptography and coding theory. In this talk, we present the study of exponential sums of boolean symmetric functions from the Experimental Mathematics perspective. In particular, we find recurrence relations they satisfy and attempt to get their exact values from these recurrences.

Joint work with: Francis N. Castro and Ivelisse Rubio.

** Mat Rogers**
Title: Mahler's measure and the WZ algorithm

The Mahler measure of an n-dimensional Laurent polynomial,
$P(x_1,...,x_n)$, is defined by
$m(P)=\int_{0}^{1}...\int_{0}^{1}\log|P(e^{2\pi i t_1},...,e^{2\pi i
t_n})|dt_1...dt_n$. There are many conjectured relations between
number-theoretic constants and Mahler measures of polynomials. In this
talk, I will show how to use the Wilf-Zeilberger algorithm to (re)prove
several formulas involving Mahler measures, which were previously proved
with algebraic K-theory. I will also mention connections with elliptic
dilogarithms.

This is joint work with Jesus Guillera.

** Eric Rowland**, Tulane University
Title: Two binomial coefficient conjectures

Much is known about binomial coefficients where primes are concerned, but considerably less is known regarding prime powers and composites. I'll discuss two conjectures in these directions, one about counting binomial coefficients modulo 16 and one about the value of binomial(n, 2p) modulo n.

** Andrew Sills**, Georgia Southern University
Title: All I Really Need To Know, I Learned From Dr. Z.

Thanks to Robert Fulghum for providing a title to parody. In order to honor Doron on his 60th birthday, I will discuss a mathematical problem that I am currently working on, deliberately highlighting the useful advice that Doron gave me during my years as his postdoc, and via his famous Opinions page.

The mathematical content of the talk is joint work with Hua Wang. The Wiener index of a graph G=(V,E), which is the sum of minimal distances between every pair of vertices in V, was introduced by H. Wiener in 1947. We consider the problem of identifying trees of maximal Wiener index among all trees that possess a given degree sequence.

** Dennis Stanton**, University of Minnesota
Title: The negative q-binomial coefficient

We give an explicit set of k-subspaces of (F_q)^n
which are counted by the negative q-binomial coefficient.
We also give a permutation enumeration interpretation of
this result. An analogous positivity conjecture for the
(-q,t)-binomial coefficients is proven, along with an
interpretation as a generating function in t over
these subspaces

This is joint work with S. Fu and V. Reiner.

** Herb Wilf**, University of Penn
Title: How to lose as little as possible

Consider a coin toss game in which each player has one coin.
Bob's coin has heads probability p and Alice's coin has heads
probability q < p. Alice chooses a positive integer n, and then each
player tosses their coin n times.The winner is the player who gets
more heads. The question is: what value of n should Alice choose?
If N(q,p) is the value of n that minimizes her expected loss, we show
that N(q,p) grows like C/(p-q), using the multivariate form of
Zeilberger's algorithm together with a lot of human mathematics, for
which we express our regrets to Doron.

Joint work with: Vittorio Addona and Stan Wagon.

** Melkamu Zeleke**, William Paterson Univ
Title: Skew diagrams and ordered trees

We use generating functions and an involution of ordered
trees to prove that among all ordered trees the ratio of the total
number of vertices to leaves is two. Ordered trees are then
enumerated by number of leaves, total path length, and number of
vertices to obtain q-analogs of Catalan numbers. The results on
ordered trees are then readily transferred by the skew diagrams to
help enumerate parallelogram polyominoes by their area, perimeter
and other statistics. We will also look at Carlitz's q-Catalan
numbers and some combinatorial objects enumerated by these numbers.