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Algebra, Sports, and Catalan Numbers

Thomas Koshy.

in Catalan Numbers with Applications

November 2008; p ublished online January 2009 .

Chapter. Subjects: Combinatorics and Graph Theory. 4951 words.

This chapter presents interesting applications of Catalan numbers to abstract algebra, linear algebra, and sports. In addition, it presents three interesting number patterns involving...

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The arithmetical triangle

A. W. F. EDWARDS.

in Combinatorics: Ancient and Modern

June 2013; p ublished online September 2013 .

Chapter. Subjects: Combinatorics and Graph Theory. 3715 words.

The arithmetical triangle is the most famous of all number patterns. Apparently a simple listing of the binomial coefficients, it contains the triangular and pyramidal numbers of ancient...

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THE ART OF PROBLEM PARAMETERIZATION

Rolf Niedermeier.

in Invitation to Fixed-Parameter Algorithms

February 2006; p ublished online September 2007 .

Chapter. Subjects: Combinatorics and Graph Theory. 3484 words.

This chapter sheds light on the various aspects of how problems can be studied from a parameterized point of view. Doing so, it emphasizes the importance of small parameters, guaranteed...

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Binomial Coefficients

Thomas Koshy.

in Catalan Numbers with Applications

November 2008; p ublished online January 2009 .

Chapter. Subjects: Combinatorics and Graph Theory. 3473 words.

This chapter presents a brief introduction to binomial coefficients, the cornerstone of all the discussions in the book. Among the properties discussed are Hermite's divisibility...

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Binomial Coefficients Revisited

Thomas Koshy.

in Catalan Numbers with Applications

November 2008; p ublished online January 2009 .

Chapter. Subjects: Combinatorics and Graph Theory. 4583 words.

This chapter presents additional properties of binomial coefficients and several occurrences of Catalan numbers. Vandermonde's identity is explained and the lattice point and...

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Block designs

NORMAN BIGGS and ROBIN WILSON.

in Combinatorics: Ancient and Modern

June 2013; p ublished online September 2013 .

Chapter. Subjects: Combinatorics and Graph Theory. 6585 words.

This chapter outlines the origins of design theory, with particular reference to the ‘Steiner triple systems’, the pioneering work of Thomas Kirkman, and early contributions by a number of...

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Catalan Numbers

Thomas Koshy.

in Catalan Numbers with Applications

November 2008; p ublished online January 2009 .

Chapter. Subjects: Combinatorics and Graph Theory. 14200 words.

This chapter presents Catalan numbers in an interesting, historical perspective, beginning with the well-known triangulation problem. In addition, it develops three recursive formulas for Cnnull...

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Catalan Numbers with Applications

Thomas Koshy.

November 2008; p ublished online January 2009 .

Book. Subjects: Combinatorics and Graph Theory. 304 pages.

Fibonacci and Lucas sequences are “two shining stars in the vast array of integer sequences,” and because of their ubiquitousness, tendency to appear in quite unexpected and unrelated...

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A Catalan Triangle

Thomas Koshy.

in Catalan Numbers with Applications

November 2008; p ublished online January 2009 .

Chapter. Subjects: Combinatorics and Graph Theory. 4333 words.

This chapter investigates an interesting triangular array that contains Catalan numbers and Cayley numbers.

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The Central Binomial Coefficient

Thomas Koshy.

in Catalan Numbers with Applications

November 2008; p ublished online January 2009 .

Chapter. Subjects: Combinatorics and Graph Theory. 12384 words.

This chapter focuses on the ubiquitous central binomial coefficient and its properties. It confirms that Cn is a positive integer. The chapter covers Lagrange's identity.

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