The KU Combinatorics Seminar meets on Wednesdays, 3:00--4:00 PM in Snow 408.
Please contact Jeremy Martin if you are interested in speaking.
Abstract: Much of matroid theory is somehow connected to the following question: Given a sequence (f_0, f_1, ..., f_r) of positive integers, when does there exist a matroid M such that each f_i is the number of independent sets of M. Stanley has conjectured a property of such sequences, which we prove in the special case when M is a lattice path matroid. We also prove an interesting connection between our result and the class of discrete polymatroids, and discuss some potential problems in extending the results. No knowledge of matroid theory will be assumed, although it may help to know the basic definitions beforehand.
The q,t-Catalan numbers naturally occur in the study of Macdonald
polynomials, which are an important family of multivariable orthogonal
polynomials introduced by Macdonald in 1988 with applications to a wide
variety of subjects including Hilbert schemes, harmonic analysis,
representation theory, mathematical physics, and algebraic
combinatorics. Haiman and Garsia-Haglund proved that they are
polynomials of q and t with positive coefficients.
Finding coefficients of the n-th q,t-Catalan number is equivalent to counting how many Catalan paths in the n*n square have the same statistics. We give simple upper bounds on coefficients in terms of partition numbers, and describe all coefficients which achieve the upper bounds.
Last updated Tue 11/17/09