Graduiertenkolleg: Methods for Discrete Structures

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Monday Lecture and Colloquium

Monday, June 11, 2007

Technische Universität Berlin
Fak. II, Institut für Mathematik
Str. des 17. Juni 136
10623 Berlin
room MA 041

Lecture - 14:15

Alexander Barvinok, University of Michigan Ann Arbor

Lattice points and rational functions

Sometimes long polynomials can be written as short rational functions.
The basic example is a formula for a finite (yet long) or an infinite
geomeric series, and the true extent of this phenomenon is not known,
although it includes generating functions for integer points in
polyhedra, lattice semigroups, and some other examples.
In the talk, I plan to survey known results, state open problems, and
sketch available techniques.

Colloquium - 16:00

Olivier Bernardi - Barcelona

Catalan intervals and realizers of triangulations

Dyck paths, binary trees and non-crossing partitions are well known combinatorial classes counted by the Catalan sequence. A classical lattice is associated to each of these classes: the Stanley lattice for the Dyck paths of length 2n, the Tamari lattice for the binary trees with n nodes and the Kreweras lattice for the non-crossing partition of [n].
In 2002, Bonichon exhibited a bijection between pairs of non-crossing Dyck paths (of size n) and realizers of triangulations (of size n). Since the order in the Stanley lattice corresponds to the relation of being above, the pairs of non-crossing Dyck paths can be seen as the intervals in this lattice.
In this talk, I will present a simpler description of the bijection of Bonichon. Then, I will explain how this bijection can be refined in order to obtain a bijection between intervals in the Tamari lattice and triangulations, and a bijection between intervals in the Kreweras lattice and ternary trees.
This is a joint work with Nicolas Bonichon.

Letzte Aktualisierung: 15.05.2007