Title: A generalization of balanced tableaux and matching problems with unique solutions
|University of British Columbia
In this talk, we consider families of finite sets that we call shellable and that have been characterized by Chang and Hirst and Hughes as being the families of sets that admit unique solutions to Hall's matching problem. We prove that shellable families can be characterized by using a generalized notion of hook-lengths; hook-lengths originate from the hook-length formula which is used to determine the number of standard Young tableaux on partition shapes. Then, we introduce a natural generalization of standard skew tableaux and Edelman and Greene's balanced tableau, then prove existence results about such a generalization using our characterization of shellable families. We also calculate the average number of such tableaux using a hook-length formula.