*Thomas Koshy*

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0006
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents a number of occurrences of Catalan numbers in a number of quite unexpected places and their relationships among them. They include mountain ranges (or Dyck paths), noncrossing ...
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This chapter presents a number of occurrences of Catalan numbers in a number of quite unexpected places and their relationships among them. They include mountain ranges (or Dyck paths), noncrossing handshakes at a round table, various classes of binary words, triangular arrays, and n-tuples of nonnegative integers.Less

This chapter presents a number of occurrences of Catalan numbers in a number of quite unexpected places and their relationships among them. They include mountain ranges (or Dyck paths), noncrossing handshakes at a round table, various classes of binary words, triangular arrays, and *n*-tuples of nonnegative integers.

*E.J. Janse Van Rensburg*

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780199666577
- eISBN:
- 9780191748653
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199666577.003.0005
- Subject:
- Mathematics, Mathematical Physics

Chapter 5 is an overview of Dyck, Motzkin and partially directed path models of polymer entropy. Adsorbing paths, the Temperley method and factorisation methods are discussed, including the Bethe ...
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Chapter 5 is an overview of Dyck, Motzkin and partially directed path models of polymer entropy. Adsorbing paths, the Temperley method and factorisation methods are discussed, including the Bethe ansatz and models of random walks in the square lattice. The chapter also discusses solution of directed path models by the kernel method. In addition, staircase polygons and directed path models in layered environments and in wedges are reviewed.Less

Chapter 5 is an overview of Dyck, Motzkin and partially directed path models of polymer entropy. Adsorbing paths, the Temperley method and factorisation methods are discussed, including the Bethe ansatz and models of random walks in the square lattice. The chapter also discusses solution of directed path models by the kernel method. In addition, staircase polygons and directed path models in layered environments and in wedges are reviewed.

*E.J. Janse van Rensburg*

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780199666577
- eISBN:
- 9780191748653
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199666577.001.0001
- Subject:
- Mathematics, Mathematical Physics

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the ...
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This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. The general theory of tricritical scaling is reviewed in the context of models of lattice clusters, and the existence of a thermodynamic limit in these models is discussed in general and for particular models. Mathematical approaches based on subadditive and convex functions, generating function methods and percolation theory are used to analyse models of adsorbing, collapsing and pulled walks and polygons in the hypercubic and in the hexagonal lattice. These methods show the existence of thermodynamic limits, pattern theorems, phase diagrams and critical points and give results on topological properties such as knotting and writhing in models of lattice polygons. The use of generating function methods and scaling in directed models is comprehensively reviewed in relation to scaling and phase behaviour in models of directed paths and polygons, including Dyck paths and models of convex polygons. Monte Carlo methods for the self-avoiding walk are discussed, with particular emphasis on dynamic algorithms such as the pivot and BFACF algorithms, and on kinetic growth algorithms such as the Rosenbluth algorithms and its variants, including the PERM, GARM and GAS algorithms.Less

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. The general theory of tricritical scaling is reviewed in the context of models of lattice clusters, and the existence of a thermodynamic limit in these models is discussed in general and for particular models. Mathematical approaches based on subadditive and convex functions, generating function methods and percolation theory are used to analyse models of adsorbing, collapsing and pulled walks and polygons in the hypercubic and in the hexagonal lattice. These methods show the existence of thermodynamic limits, pattern theorems, phase diagrams and critical points and give results on topological properties such as knotting and writhing in models of lattice polygons. The use of generating function methods and scaling in directed models is comprehensively reviewed in relation to scaling and phase behaviour in models of directed paths and polygons, including Dyck paths and models of convex polygons. Monte Carlo methods for the self-avoiding walk are discussed, with particular emphasis on dynamic algorithms such as the pivot and BFACF algorithms, and on kinetic growth algorithms such as the Rosenbluth algorithms and its variants, including the PERM, GARM and GAS algorithms.