Combinatorics - The Rota Way

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. Written by two of his 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, Möbius 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.