Gian-Carlo Rota (1932-1999) was one of the most original and colorful mathematicians of the twentieth century. His work revealed the underlying algebraic structures in combinatorics and inspired a new area of algebraic combinatorics. His graduate courses at the Massachusetts Institute of Technology on the foundations of combinatorics influenced generations of students.
This book, written by two of his former students, is based on notes from his graduate courses and personal discussions with him. Topics include partially ordered sets, partitions and entropy, matching theory, Moebius functions, Sperner theory, distributive, geometric, linear, and modular lattices, valuation rings, umbral calculus, symmetric functions, Baxter algebras, and location of zeros of polynomials. Many exercises and research problems are included. In addition, unexplored areas of possible research are discussed.
Students and researchers in combinatorics and related areas of mathematics will find much to interest them in this book.
Comments and Corrections (It will be updated regularly.)