Abstract: The Matrix-Tree Theorem can be used to give a simple formula for the number of spanning trees of the n-dimensional hypercube. However, no bijective proof of this formula is known. I'll discuss a more refined version of the formula, proven by Vic Reiner and myself, which we hope will point the way to a bijection.
Abstract: Degree sequences of graphs have been thoroughly studied. For example, there are many simple characterizations of when an integer sequence is the degree sequence of a graph and of graphs with "extremal" degree sequences. Notions of generalized degree sequences for higher dimensional simplicial complexes are not as well investigated. I will talk about work in progress on understanding these degree sequences and those classes of complexes which exhibit analogous extremal behavior. This is joint work with Uri Peled and Amitava Bhattacharya.
Abstract: We are interested in Stanley's question of whether the chromatic symmetric function distinguishes nonisomorphic trees. Instead of working with trees in general, we simplify to the problem of distinguishing between certain types of caterpillars. Then, using a bijection between ribbon diagrams and caterpillars, we look at what properties of the ribbons appear in the corresponding chromatic symmetric function.
Abstract: The partially asymmetric excluion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. Particles may enter the system from the left with probability alpha, and exit from the right with probability beta. The model is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. In this talk, we will describe a surprising connection between the PASEP model and the combinatorics of certain 0-1 tableaux called permutation tableaux. Namely, we prove that in the long time limit, the probability that the PASEP is in a particular configuration tau is a generating function for permutation tableaux of shape lambda(tau), enumerated according to three statistics. The tableaux in question come from total positivity on the Grassmannian (via work of Postnikov).
Last updated Wed 8/23/06 10:00 PM