Combinatorics seminar
Combinatorics Seminar  Fall 2014
Tuesday, 1011am
Room 209, Goldsmith building
The Combinatorics Seminar is an introductory seminar for combinatorics. The talk should be accesible to first year graduate students.

Tuesday September 30
Speaker: Yan Zhuang
Title: Counting Permutations by Alternating Descents
Abstract: We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. We give two proofs for the generating function. The first proof uses a system of differential equations whose solution gives the generating function. For the second proof, we develop a new method for counting permutations by alternating descents using noncommutative symmetric functions. This is joint work with Ira Gessel.

Tuesday October 7
Speaker: Ira Gessel
Title: Exponential generating functions modulo p
Abstract:
The exponential generating function for a sequence a_n is the formal power series in which the coefficient of x^n/n! is a_n. Exponential generating functions are useful in counting "labeled objects" such as permutations, partitions, and graphs. We would like to find congruences to prime moduli for these sequences; for example, the Bell numbers and the tangent and secant numbers.
Formal power series that are exponential generating functions for sequences of integers are called Hurwitz series. Hurwitz series form a ring, and they are also closed under composition. We can obtain the congruences that we want by studying the quotient ring of Hurwitz series modulo a prime p. The structure is easy to describe, but more important are derivations and their chain rules. The derivative D with respect to x is a derivation, but so is D^m whenever m is a power of p. Using these derivations we can prove that the set of Hurwitz series whose coefficients are periodic (mod p) is closed under the usual operations (including composition).
Previous semesters:
Spring 2014,
Fall 2013.