Trjitzinsky Memorial Lectures

Herbert Wilf
University of Pennsylvnia

October 13-15, 2003

Monday, October 13, 4 p.m., 165 Everitt
1. Recounting the Rationals
We all know that the rationals are countable. Here we will show that there is a single sequence b(0), b(1), b(2), b(3), ... of positive integers that has both of the following properties:
(a) The sequence b(0)/b(1), b(1)/b(2), b(2)/b(3), ... is a list of every positive rational number, each occurring once and only once, and in lowest terms automatically (i.e., b(n) and b(n+1) are always relatively prime).
(b) The numbers b(n) are the solution of an easy-to-state counting problem. That is, b(n) is the number of ways of doing something, the "something" being a fairly standard combinatorial activity.
This result was found by Neil Calkin and myself in 2001, and we will discuss some of the generalizations and consequences that have been found since then.

Tuesday, October 14, 4 p.m., 314 Altgeld Hall
A reception will be held following this lecture in 321 Altgeld Hall.
2. The WZ Method
This method, for automatically finding when hypergeometric sums can be expressed in simple closed form, finding that closed form when it exists, and proving that they cannot be so expressed when that is the case, is now roughly ten years old. We will give an overview of it, along with some questions that remain unanswered.

Wednesday, October 15, 4 p.m., 245 Altgeld Hall
3. Permutation Patterns
If tau is a given permutations of k letters, then tau may or may not occur as a pattern in some permutation of n > k letters. A pattern is a subsequence of the values of the larger permutation whose size ranks correspond to the given pattern tau. For example the pattern (132) occurs in the permutation (14532), for instance in the subsequence 153, while the permutation (32145) avoids the pattern (132) altogether. The study of these patterns has been extremely active in the past decade. I'll give a survey of some results and of some of the outstanding unsolved problems in the field, including a description of some new results of Emeric Deutsch, A. J. Hildebrand, and myself, on limiting distributions.

 
History of Trjitzinsky Lectures

Department Weekly Calendar |  Mathematics Seminars |  Colloquia Schedule |  Conferences |  Calendar Archives