# A Tribute to Herbert S. Wilf

### in celebration of his 65th birthday

On June 12 - 15, 1996, a Conference on Combinatorics and Graph Theory was held at the University of Pennsylvania to honor Herb Wilf and his many contributions to mathematics and mathematicians; on that occasion we also celebrated his 65th birthday. This special issue of the Electronic Journal of Combinatorics is a collection of papers by students, coauthors and friends who were among the 186 participants who gathered to pay tribute to Herb Wilf -- an outstanding mathematician and a truly wonderful teacher.

The mathematics of Herb Wilf can be divided into three main phases. First was numerical analysis where Herb did his Ph. D. dissertation (under Herbert Robbins at Columbia University in 1958) and his first papers were written. Next was complex analysis and the theory of inequalities, in particular, Hilbert's inequalities restricted to n variables. There was a cluster of papers by Herb, some with de Bruijn and some with Harold Widom. The main area of Herb Wilf is combinatorics. In 1965, Gian-Carlo Rota came to the University of Pennsylvania to give a colloquium talk on his then-recent work on Mobius functions and their role in combinatorics. Herb recalled, "That talk was so brilliant and so beautiful that it lifted me right out of my chair and made me a combinatorialist on the spot." The next year, Herb had a sabbatical leave in London and he spent his time in the library at Imperial College going through every paper in combinatorics since the creation of the world. By the end of the year, Herb had proved that the chromatic number of a graph is at most 1 plus the largest eigenvalue of its adjacency matrix with equality only as in Brooks' theorem, thus making his debut in combinatorics. Since then, Herb has had a profound influence in numerous areas of combinatorics, analysis, graph theory, discrete algorithms and, in particular, in generating functions through his pioneering work, with Doron Zeilberger, in computerized proofs of combinatorial identities.

Long before most researchers and educators, Herb Wilf recognized the important connections between computers and mathematics. As early as his undergraduate years at MIT (1951), he took summer jobs at IBM in Manhattan where there was a mainframe IBM 701 designed by von Neumann. With Tony Ralston, he wrote a program implementing a multistage linear regression analysis, following a method developed by Mike Efroymson, then of Esso Corp. The program was used by Esso (now Exxon) to control the quality of refined gasoline. Together, Wilf and Ralston edited the book Mathematical Methods for Digital Computers, and two more volumes followed a few years later. In Herb's graduate years, in order to support his growing family, he took partial and full time jobs as the head of computing section with a small staff of one (himself) at Fairchild Engine Division and also at Nuclear Development Associates. His work there included calculating eigenvalues of a proposed jet engine design on the IBM CPC, generating random objects and using Monte Carlo methods. After he received his Ph.D. and became a professor, the computer continued to play a key role in both his research and teaching. He and Albert Nijenhuis masterfully integrated FORTRAN with combinatorics as seen in their book Combinatorial Algorithms. His celebrated work with Petkovsek and Zeilberger concerns computer-automated proofs which has expanded the notion of mathematical proofs and resulted in a recent book A=B .

Herb Wilf has co-founded two major journals, Journal of Algorithms in 1980 with Donald Knuth, and The Electronic Journal of Combinatorics in 1994 with Neil Calkin, for which he serves as an Editor-in-Chief. During his tenure as the editor for the American Mathematical Monthly (1987-1992), in addition to his editorial duties, he wrote a series of illuminating expository articles.

To this date, Herb Wilf has had twenty Ph.D. students, 36 coauthors, eleven books, and over 130 research papers. The clickable images here lead to larger images, designed by Stan Wagon, that display the full list of students and coauthors at the time of the conference.

Those of us whose lives have been touched by Herb's extraordinary generosity and talent in writing and teaching are pleased to take this opportunity to express our sincere thanks.

Fan Chung, special issue editor and conference co-chair

Joan Hutchinson, conference co-chair

Graphic design by Stan Wagon