#
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**

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Alexander Barvinok, University of Michigan Ann Arbor

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Lattice points and rational functions

*Abstract:*

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**

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Olivier Bernardi - Barcelona

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Catalan intervals and realizers of triangulations

*Abstract:*

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