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.
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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.
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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