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Affine Fixed Point Buildings

Bernhard M¨uhlherr, Holger P. Petersson and Richard M. Weiss.

in Descent in Buildings (AM-190)

September 2015; p ublished online October 2017 .

Chapter. Subjects: Combinatorics and Graph Theory. 4733 words.

This chapter shows that if Ξ‎ is an affine building and Γ‎ is a finite descent group of Ξ‎, then Γ‎ is a descent group of Ξ‎ and (Ξ‎) is congruent to (Ξ‎). Ξ‎Γ‎ and Ξ‎ can be viewed as...

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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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Arithmetic of Unicritical Polynomial Maps

John Milnor.

Edited by Araceli Bonifant, Mikhail Lyubich and Scott Sutherland.

in Frontiers in Complex Dynamics

March 2014; p ublished online October 2017 .

Chapter. Subjects: Combinatorics and Graph Theory. 3213 words.

This chapter studies complex polynomials with only one critical point, relating arithmetic properties of the coefficients to those of periodic orbits and their multipliers and external...

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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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Bifurcation currents and equidistribution in parameter space

Romain Dujardin.

Edited by Araceli Bonifant, Mikhail Lyubich and Scott Sutherland.

in Frontiers in Complex Dynamics

March 2014; p ublished online October 2017 .

Chapter. Subjects: Combinatorics and Graph Theory. 25691 words.

This chapter reviews the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P¹ or subgroups of...

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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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Buildings

Bernhard M¨uhlherr, Holger P. Petersson and Richard M. Weiss.

in Descent in Buildings (AM-190)

September 2015; p ublished online October 2017 .

Chapter. Subjects: Combinatorics and Graph Theory. 4880 words.

This chapter assembles a few standard definitions, fixes some notation, and reviews a few of the results about buildings and Moufang polygons. It also summarizes the basic facts about...

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