Gian-Carlo Rota was one of the most original and colourful mathematicians of the 20th century. His work on the foundations of combinatorics focused on the algebraic structures that lie behind diverse combinatorial areas, and created a new area of algebraic combinatorics. Written by two of his former students, this book is based on notes from his influential graduate courses and on face-to-face discussions. 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. A must-have for all students and researchers in combinatorics and related areas. Gian-carlo Rota Was One Of The Most Original And Colourful Mathematicians Of The 20th Century. Written By Two Of His Former Students, This Book Is Based On Notes From His Influential Graduate Courses And On Face-to-face Discussions. Sets, Functions, And Relations. Sets, Valuations, And Boolean Algebras -- Partially Ordered Sets -- Lattices -- Functions, Partitions, And Entropy -- Relations -- Matching Theory. What Is Matching Theory? -- The Marriage Theorem -- Free And Incidence Matrices -- Submodular Functions And Independent Matchings -- Rado's Theorem On Subrelations -- Doubly Stochastic Matrices -- The Gale-ryser Theorem -- Matching Theory In Higher Dimensions -- Partially Ordered Sets And Lattices. Möbius Functions -- Chains And Antichains -- Sperner Theory -- Modular And Linear Lattices -- Finite Modular And Geometric Lattices -- Valuation Rings And Möbius Algebras -- Generating Functions And The Umbral Calculus. Generating Functions -- Dt Elementary Umbral Calculus -- Polynomial Sequences Of Binomial Type -- Sheffer Sequences -- Umbral Composition And Connection Matrices -- The Riemann Zeta Function -- Symmetric Functions And Baxter Algebras. Symmetric Functions -- Distribution, Occupancy, And The Partition Lattice -- Enumeration Under A Group Action -- Baxter Operators -- Free Baxter Algebras -- Identities In Baxter Algebras -- Symmetric Functions Over Finite Fields -- Historical Remarks And Further Reading -- Determinants, Matrices, And Polynomials. Polynomials -- Apolarity -- Grace's Theorem -- Multiplier Sequences -- Totally Positive Matrices -- Exterior Algebras And Compound Matrices -- Eigenvalues Of Totally Positive Matrices -- Variation Decreasing Matrices -- Pólya Frequency Sequences -- Selected Solutions. Joseph P.s. Kung, Gian-carlo Rota, Catherine H. Yan. Includes Bibliographical References (p. 369-388) And Index. 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.
Compiled and edited by two of Gian-Carlo Rota's students, this book is based on notes from his influential combinatorics courses.
