The Unity of Combinatorics

ISBN-13: 9781470465094

Ezra Brown;Richard K. Guy

Opis

Combinatorics, or the art and science of counting, is a vibrant and active area of pure mathematical research with many applications. The Unity of Combinatorics succeeds in showing that the many facets of combinatorics are not merely isolated instances of clever tricks but that they have numerous connections and threads weaving them together to form a beautifully patterned tapestry of ideas. Topics include combinatorial designs, combinatorial games, matroids, difference sets, Fibonacci numbers, finite geometries, Pascal's triangle, Penrose tilings, error-correcting codes, and many others. Anyone with an interest in mathematics, professional or recreational, will be sure to find this book both enlightening and enjoyable. Few mathematicians have been as active in this area as Richard Guy, now in his eighth decade of mathematical productivity. Guy is the author of over 300 papers and twelve books in geometry, number theory, graph theory, and combinatorics. In addition to being a life-long number-theorist and combinatorialist, Guy's co-author, Ezra Brown, is a multi-award-winning expository writer. Together, Guy and Brown have produced a book that, in the spirit of the founding words of the Carus book series, is accessible ""not only to mathematicians but to scientific workers and others with a modest mathematical background."

Spis treści

• Introduction
• Blocks, sequences, bow ties, and worms
• Combinatorial games
• Fibonacci, Pascal, and Catalan
• Catwalks, sandsteps, and Pascal pyramids
• Unique rook circuits
• Sums, colorings, squared squares, and packings
• Difference sets and combinatorial designs
• Geometric connections
• The groups $PSL(2,7)$ and $GL(3,2)$ and why they are isomorphic
• Incidence matrices, codes, and sphere packings
• Kirkman's Schoolgirls, fields, spreads, and hats
• $(7,3,1)$ and combinatorics
• $(7,3,1)$ and normed algebras
• $(7,3,1)$ and matroids
• Coin turning games and Mock Turtles
• The $(11,5,2)$ biplane, codes, designs, and groups
• Rick's Tricky Six Puzzle: More than meets the eye
• $S(5,8,24)$
• The miracle octad generator
• Bibliography
• Index