Combinatorics. The Rota way. (English) Zbl 1159.05002

Kung, Joseph P. S.; Rota, Gian-Carlo; Yan, Catherine H.

Cambridge Mathematical Library. Cambridge: Cambridge University Press (ISBN 978-0-521-88389-4/hbk; 978-0-521-73794-4/pbk). xii, 396 p. (2009).

Gian-Carlo Rota’s 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. This book was written by two of his former students, based on notes from his courses and on personal discussions with him.
Chapter 1 (Sets, functions, and relations) is about sets, functions, relations, valuations, and entropy. Chapter 2 (Matching theory) includes topics on free and incidence matrices, submodular functions and independent matchings, Rado’s theorem on subrelations, doubly stochastic matrices, the Gale-Ryser theorem and matching theory in higher dimensions. The third chapter (Partially ordered sets and lattices) offers a mixture of topics in partially ordered sets: Möbius functions, Sperner theory, modular, linear and geometric lattices, valuation rings and Möbius algebras, Dilworth’ s chain partition theorem. Chapter 4 (Generating functions and the umbral calculus) includes sections about generating functions, elementary umbral calculus, polynomial sequences of binomial type, Sheffer sequences, umbral composition and connection matrices, the Riemann zeta function. Chapter 5 (Symmetric functions and Baxter algebras) is about symmetric functions defined by distribution and occupancy and applied to the study of Baxter algebras. This chapter ends with a section on symmetric functions over finite fields. The sixth chapter (Determinants, matrices, and polynomials) is on polynomials and their zeros. It includes apolarity, Grace’ s theorem, multiplier sequences, totally positive matrices and their eigenvalues, exterior algebras and compound matrices, variation decreasing matrices and Pólya frequency sequences.
Each chapter ends with exercises followed by historical remarks and further reading. A section is devoted to selected solutions of the exercises. Also there is a comprehensive bibliography and an index. A valuable book addressed to all students and researchers in combinatorics and related areas.

Reviewer: Ioan Tomescu (Bucureşti)

MSC:

05-02	Research exposition (monographs, survey articles) pertaining to combinatorics
06-02	Research exposition (monographs, survey articles) pertaining to ordered structures
05Axx	Enumerative combinatorics
05Dxx	Extremal combinatorics
06Axx	Ordered sets
06Bxx	Lattices

Keywords:

combinatorics; matching theory; doubly stochastic matrix; Gale-Ryser theorem; partially ordered set; Möbius function; Sperner theory; finite modular and geometric lattice; Möbius algebra; generating function; umbral calculus; Sheffer sequence; symmetric function; Baxter algebra; totally positive matrix; exterior algebra; Pólya frequency sequences