Intendierte Lernergebnisse
After successfully completing this course, students will know the main results concerning combinatorial reciprocity: They can explain and prove interesting connections between counting sequences of different combinatorial objects. Moreover, they can apply combinatorial and geometric methods to derive combinatorial reciprocity results, and they will also be able to verify special cases of reciprocity results using a computer algebra system.
Lehrmethodik
Lecture with interactive elements
Inhalt/e
The content of this course lies in the interplay between enumerative and geometric combinatorics. The main theme will be the study of a fascinating phenomenon known as combinatorial reciprocity, which relates the enumeration of two families of combinatorial objects through the evaluation of a polynomial at positive and negative integers. The main objective is to develop combinatorial and geometric tools to derive and explain combinatorial reciprocities. In the process, we will learn aboutpartially ordered sets and order polynomials,colorings of graphs,acyclic orientations and flows, andcounting regions of hyperplane arrangements.We will also see how the geometry of convex polytopes and polyhedra can beautifully explain many of the combinatorial results presented in this course.
Erwartete Vorkenntnisse
Basic knowledge in combinatorics and graph theory as taught in the course „Kombinatorische Strukturen“, programming skills as taught in the course „Computermathematik“, and profound knowledge of proof techniques are assumed.
Literatur
Matthias Beck and Raman Sanyal, Combinatorial Reciprocity Theorems: An Invitation To Enumerative Geometric Combinatorics, Graduate Studies in Mathematics, 195. American Mathematical Society, 2018.Richard Stanley, Combinatorial reciprocity theorems, Advances in Mathematics 14 (1974), 194–253.Günter Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995.