# Lectures on bijections at IPAM

**Workshop:**
IPAM Combinatorics tutorials.

**Speaker:** Igor Pak, UCLA.

**Date and time: **September 9-11, 2009.

**Place:** IPAM.

## Lecture 1. Tree bijections.

Cayley formula on the number of spanning trees in a complete
graph is one of the most celebrated results in combinatorics.
The number and variety of bijective proofs of this formula is
truly astonishing, many of them having important generalizations
and interesting corollaries. I will survey a number of such tree
bijections, emphasizing probabilistic connections and applications.
In this, as in the other lectures, no previous familiarity with
the subject is assumed.

Download .pdf file of the lecture.

## Lecture 2. Partition bijections.

The study of partition identities goes back to the works of Euler,
Gauss and Jacobi, and have been flourishing ever since. In the late
19th century, J.J. Sylvester singlehandedly revolutionized the field
by introducing a "constructive approach" of proving partition
identities with partition bijection, and showing how to apply
it in a number of important cases. As we understand now, the
highly positive outlook on the power of partition bijections
was destroyed by Ramanujan who introduced literally hundreds
of new partitions identities, many of which were (and some still are)
difficult to prove even analytically. In this lecture I will give
a broad survey of partition bijection proving various
pre- and post-Ramanujan partition identities. At the end I will
also discuss the complexity of O'Hara's algorithm, due to
Konvalinka and myself.

Download .pdf file of the lecture.

## Lecture 3. Young tableaux bijections.

Young tableaux were introduced by Alfred Young on the verge of
the 20-th century, in the context of covariants of the symmetric
group. Since then, they have appeared in a number of contexts
ranging from representation theory to discrete probability,
from enumerative algebraic geometry to statistical physics.
In this lecture I will review some remarkable Young tableaux
bijections developed over the past century.

Download .pdf file of the lecture.

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*Last updated 9/12/2009*