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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2004-2005 MATHEMATICS COLLOQUIUM SERIES 01/20/05 Colloquium Dr. Dan Warren Department of Mathematics University of Florida Optimizing the Packing Behavior of Layered Permutation Patterns Abstract:   Originating from questions related to stack-sortability, the subject of pattern avoidance in permutations has for the past few ears been a subject of intense study in Combinatorics. In 1992, the idea of pattern packing was proposed by H. Wilf as an opposite question to pattern avoidance. A permutation is called layered if it can be decomposed into blocks of consecutive entries which are descending within the blocks and ascending between blocks, e.g., 32154876. I will define the packing density of a permutation, discuss the impact of layering on the computation of packing density, and compute the packing densities of a few families of permutations. Mr. Warren expects to receive the Ph.D. in Mathematics from the University of Florida in May, 2005. He received his BS in Mathematics from the University of Georgia (summa cum laude).