Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson

Abstract

In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, Combinatorial Enumeration, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collaborative work.Separately-but most notably together-Ian Goulden and David Jackson (pictured in Figure 2, Page 17) have made a major impact in the fields of algebraic and combinatorial enumeration. Their contributions run the gamut from writing a groundbreaking textbook and authoritative reference (Combinatorial Enumeration, Wiley, 1983 [45]), to co-founding with Chris Godsil a well-regarded journal (Journal of Algebraic Combinatorics, founded in 1992), to publishing over 50 research papers together in their collaboration lasting over 40 years. They have also been invited speakers of the main conference in the field of algebraic combinatorics Formal Power Series and Algebraic Combinatorics (FPSAC) three times (Goulden in 1995, and Jackson in 1994 [34]. Along the way they have supervised numerous graduate students and mentored many more undergraduates and colleagues. This survey article explores some of their key results.David Jackson graduated from the University of Cambridge in 1970 with a Ph.D. in mathematics. Within three years he had joined the Faculty of Mathematics at the University of Waterloo, where he would spend the rest of his professorial career. Ian Goulden was one of Jackson's earliest Ph.D. students, graduating from the University of Waterloo in 1979 with a Ph.D. in Statistics. Shortly thereafter Goulden began his professional career in the Department of Combinatorics and Optimization at the University of Waterloo.All of their work spotlights a respect for rigour, a delight in tackling challenging problems, and a drive to understand and explain complex ideas at the interface between algebra and combinatorics. It's a challenging area-the practitioner must be accomplished in two domains of mathematics-but it's a rewarding one. Goulden and Jackson set a high bar. Time and again they have set in motion (or thrown down the gauntlet in) various areas of research. They are responsible for a number of exciting conjectures, some of which we highlight throughout this survey.