## 2nd Algorithmic and Enumerative CombinatoricsSummer School 2015

### Invited Speakers

• Matthias Beck (San Francisco State University, U.S.A.)

Rational Generating Functions of Order Cones and Applications
Absract: To a given finite partially ordered set (Π ,  ≼ ), we associate its order cone
KΠ :  = {x ∈ R ≥ 0Π :  j ≼ k  ⇒  xj ≤ xk}.
Abstractly, KΠ  is the set of all poset homomorphisms (Π ,  ≼ ) → (R ≥ 0,  ≤ ), but we may think of KΠ  simply as a geometric realization of (Π ,  ≼ ) in d-dimensional Euclidean space, where d:  = ∣Π ∣. It turns out that the geometry of KΠ  is quite rich, and one of our goals is to study the polyhedral geometry that underlies order cones. The arithmetic of an order cone is captured in the generating function
σΠ (z1, …, zd):  = ∑ (m1, …, md) ∈ (KΠ  ∩ Zd) z1m1zdmd.
As we will see, this generating function evaluates to a rational function in z1, …, zd, and the study of these rational generating functiond is our second main goal.

There are many applications of order cones, most prominently in partition analysis and permutation statistics. In fact, the above generating function goes back to Stanley’s P-partitions, and so our third goal is to illustrate the rich connections between certain ideas from number theory, combinatorics, and geometry.

Note: The lectures will concentrate on Chapters 4 and 6 of the brandnew book Combinatorial Reciprocity Theorems: An Invitation To Enumerative Geometric Combinatorics .

• Igor Pak (University of California, Los Angeles, U.S.A.)

Computability and Enumeration
Abstract: Classical combinatorial objects tend to be nice and easy to enumerate. They have nice asymptotics and their generating functions satisfy various analytic properties. In this series of lectures I will concentrate on large classes of combinatorial objects, such as tilings, words and pattern avoiding permutations. We show that such classes are rich enough to contain combinatorial objects which are computationally hard to enumerate. Notable examples include our (negative) solution of Kontsevich's problem for counting trivial words in linear groups, and our recent disproof of the Noonan-Zeilberger Conjecture.

• Mark van Hoeij (Florida State University, U.S.A.)

Combinatorial Problems and Hypergeometric Solutions of Linear Differential Equations
Abstract: If the generating function of an integer sequence is convergent and satisfies a second order linear differential equation, then it conjecturally can be written in terms of hypergeometric functions. However, combinatorics not only leads to differential equations with closed form solutions, it can also help to find such solutions. Dessins d'enfants, and braid orbits of near-dessins, are combinatorial objects that correspond to functions via the Riemann Existence Theorem. The lectures will explain these objects, how to tabulate them, and show how this can be used to solve linear differential equations.