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Pascal' Triangle and Catalan Numbers

Thomas Koshy

in Catalan Numbers with Applications

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.003.0012
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter shows the various ways Catalan numbers can be extracted from Pascal's triangle. It includes discussion of nonisomorphic groups, Catalan polynomials, Touchard's recursive formula, and ... More


EXPANDING THE TUTTE POLYNOMIAL OF A MATROID OVER THE INDEPENDENT SETS

Koko Kalambay Kayibi

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0011
Subject:
Mathematics, Probability / Statistics

This chapter provides direct combinatorial proof of an expansion of the Tutte polynomial by independent sets of the matroid. Another expansion of the Tutte polynomial is presented in terms of ... More


ORBIT COUNTING AND THE TUTTE POLYNOMIAL

Peter J. Cameron

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0001
Subject:
Mathematics, Probability / Statistics

This chapter summarizes the various attempts to extend the Tutte polynomial of a matroid to a polynomial which counts orbits of a group on various sets of objects that the usual Tutte polynomial ... More


COMPLEXITY OF GRAPH POLYNOMIALS

Steven D. Noble

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0013
Subject:
Mathematics, Probability / Statistics

This chapter examines the complexity of evaluating graph polynomials, related to the Tutte polynomial, for various classes of matroids. It begins with a short introduction to matroids, complexity, ... More


TUTTE-WHITNEY POLYNOMIALS: SOME HISTORY AND GENERALIZATIONS

Graham E. Farr

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0003
Subject:
Mathematics, Probability / Statistics

The Tutte-Whitney polynomials play a key role in the study of counting problems on graphs, and have close connections with statistical mechanics and knot theory. This chapter briefly reviews their ... More


APPROXIMATING THE TUTTE POLYNOMIAL

Mark Jerrum

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0009
Subject:
Mathematics, Probability / Statistics

This chapter examines some algorithmic problems associated with matroids. It focuses on determining a ‘fully polynomial randomized approximation scheme’ or ‘FPRAS’. First, the problem of counting ... More


Finding the Roots of Algebraic Equations

Victor J. Katz and Karen Hunger Parshall

in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691149059
eISBN:
9781400850525
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691149059.003.0011
Subject:
Mathematics, History of Mathematics

This chapter highlights some of the struggles to solve fifth-degree polynomials as well as the development of the idea of a group and its application to answering the question of solvability of ... More


Sonine Polynomials Expansion of The Velocity Distribution Function

Nikolai V. Brilliantov and Thorsten Pöschel

in Kinetic Theory of Granular Gases

Published in print:
2004
Published Online:
January 2010
ISBN:
9780198530381
eISBN:
9780191713057
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198530381.003.0007
Subject:
Physics, Condensed Matter Physics / Materials

The velocity distribution of function of a granular gas is different from the Maxwell distribution. It may be represented in the form of a Sonine polynomials expansion. This chapter shows that the ... More


Thermoelastic Polynomials

Józef Ignaczak and Martin Ostoja‐Starzewski

in Thermoelasticity with Finite Wave Speeds

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199541645
eISBN:
9780191716164
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199541645.003.0009
Subject:
Mathematics, Applied Mathematics, Mathematical Physics

This chapter begins from the observation that the fundamental solutions of the G‐L theory may be determined with the help of polynomial sequences on the time axis, the so‐called polynomials of ... More


Knowability

Ben Brubaker, Daniel Bump, and Solomon Friedberg

in Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory (AM-175)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691150659
eISBN:
9781400838998
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691150659.003.0012
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the ... More


Sylvester denumerant

J. L. Ramírez Alfonsín

in The Diophantine Frobenius Problem

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198568209
eISBN:
9780191718229
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568209.003.0004
Subject:
Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics

In 1857, while investigating the partition number function, J. J. Sylvester defined the function d(m; a1, . . . , an), called the denumerant, as the number of nonnegative integer representations of m ... More


TESTING FOR THE EXISTENCE OF HOMOMORPHISMS

Pavol Hell and Jaroslav Nešetřil

in Graphs and Homomorphisms

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198528173
eISBN:
9780191713644
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528173.003.0005
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter explores the algorithmic aspects of graph homomorphisms and of similar partition problems. The highlights include the dichotomy classification of graph homomorphisms to a fixed target ... More


Harmonic morphisms defined by polynomials

Paul Baird and John C. Wood

in Harmonic Morphisms Between Riemannian Manifolds

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198503620
eISBN:
9780191708435
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198503620.003.0005
Subject:
Mathematics, Pure Mathematics

This chapter establishes the Liouville-type theorem which asserts that any entire (i.e., globally defined) harmonic morphism from a Euclidean space of arbitrary dimension to a Euclidean space of ... More


SL(n) Examples with Slightly Composite n

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0023
Subject:
Mathematics, Number Theory

This chapter continues to study the object N of Theorem 21.1. Thus, k is a finite field of characteristic p, ψ‎ a nontrivial additive character of k, f(x) = f(x)=∑i=−baAixi∈k[x,1/x]Aᵢxⁱ ɛ k[x, 1/x] ... More


Other SL(n) Examples

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0024
Subject:
Mathematics, Number Theory

This chapter fixes an integer n ≤ 3 which is not a power of the characteristic p, and a monic polynomial f(x) ɛ k[x] of degree n, f(x)=∑i=0nAixi, An=1.


FOURIER ANALYSIS ON FINITE ABELIAN GROUPS: SOME GRAPHICAL APPLICATIONS

Andrew Goodall

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0007
Subject:
Mathematics, Probability / Statistics

This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in ... More


FLOWS AND FERROMAGNETS

Geoffrey Grimmett

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0008
Subject:
Mathematics, Probability / Statistics

The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety ... More


INTERACTIVE PROGRAMS AND WEAKLY FINAL COALGEBRAS IN DEPENDENT TYPE THEORY

Peter Hancock and Anton Setzer

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566519
eISBN:
9780191713927
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566519.003.0007
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter considers the representation of interactive programs in dependent type theory and bases it on the monadic I/O used in Haskell. Two versions are described: in the first the interface with ... More


CONNECTIONS TO APPROXIMATION ALGORITHMS

Rolf Niedermeier

in Invitation to Fixed-Parameter Algorithms

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198566076
eISBN:
9780191713910
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566076.003.0014
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter summarizes the currently known connections between fixed-parameter and polynomial-time approximation algorithmics. Important topics in this context are lower bound and impossibility ... More


THE CONTRIBUTIONS OF DOMINIC WELSH TO MATROID THEORY

James Oxley

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0015
Subject:
Mathematics, Probability / Statistics

Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh's work in and ... More


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