Combinatorics: The Rota Way

Written by two of Gian-Carlo Rota's former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Moebius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exercises and research problems are included, and unexplored areas of possible research are discussed. This book should be on the shelf of all students and researchers in combinatorics and related areas.

Gian-Carlo Rota was one of the most original and colorful mathematicians of the twentieth century. His work on the foundations of combinatorics focused on revealing the algebraic structures that lie behind diverse combinatorial areas and created a new area of algebraic combinatorics. His graduate courses influenced generations of students. Joseph P.S. Kung is a professor of mathematics at the University of North Texas. He is currently an editor-in-chief of Advances in Applied Mathematics. Gian-Carlo Rota (1932-1999) was a professor of applied mathematics and natural philosophy at the Massachusetts Institute of Technology. He was a member of the National Academy of Science. He was awarded the 1988 Steele Prize of the American Mathematical Society for his 1964 paper "On the Foundations of Combinatorial Theory I. Theory of Moumlet;bius Functions." He was a founding editor of Journal of Combinatorial Theory. Catherine H. Yan is a professor of mathematics at Texas AM University. Prior to that, she was a Courant Instructor at New York University and a Sloan Fellow.