Combinatorics Seminar - Fall 2014
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
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).