Ward Resolves One Of Herbert Wilf's Unsolved Problems

May,  2010


In March 2010, Dr. Mark Daniel Ward, Assistant Professor in Statistics, resolved the third of eight Unsolved Problems posed by Herbert Wilf. Wilf is a well known expert in several branches of combinatorics; he is the Thomas A. Scott Professor of Mathematics Emeritus at the University of Pennsylvania. The problems are posted on his webpage to inspire discussion and (hopefully) eventual solutions of these very challenging questions in combinatorics. The third problem, now resolved by Ward, involves fractions that are closer and closer approximations of the transcendental number pi. Fractions for these approximations are written as coefficients of a generating function; the question posed by Wilf is about the first order asymptotic behavior of these approximations, namely, how quickly do these approximations improve? Ward's solution relies on complex analysis (a separate branch of mathematics from combinatorics) to precisely determine the asymptotic behavior of these approximations. Such connections between different areas of mathematics are often needed to resolve such challenging questions. Ward's paper on the solution has been accepted for the 21st International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms. The papers of the proceedings from the meeting will be published in a special volume of Discrete Mathematics and Theoretical Computer Science later this year.