## Generalized 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/9780195334548.003.0017
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter generalizes Catalan numbers and introduces Fuss numbers.

## Divisibility Properties

*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/9780195334548.003.0013
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates several divisibility properties of shows Catalan numbers. They include their parity and primality. Mersenne numbers are also described.

## A Catalan Triangle

*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/9780195334548.003.0014
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

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

## Partitions 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/9780195334548.003.0010
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates several occurrences of Catalan numbers in the theory of partitioning. Noncrossing partition is explained in detail.

## A Family of Binary Words

*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/9780195334548.003.0015
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter generalizes the family of binary numbers studied in Chapter 5.

## Notes on memory (1945)

*Alan M. Turing*

### in Alan Turing's Automatic Computing Engine: The Master Codebreaker's Struggle to build the Modern Computer

- Published in print:
- 2005
- Published Online:
- January 2008
- ISBN:
- 9780198565932
- eISBN:
- 9780191714016
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198565932.003.0022
- Subject:
- Mathematics, History of Mathematics

This chapter presents fragments from Turing's draft of ‘Proposed Electronic Calculator’. This material is of interest chiefly because of its remarks concerning the universal machine of Turing's 1936 ... More

## Science in the Looking Glass: What Do Scientists Really Know?

*E. Brian Davies*

- Published in print:
- 2007
- Published Online:
- September 2008
- ISBN:
- 9780199219186
- eISBN:
- 9780191711695
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219186.001.0001
- Subject:
- Physics, History of Physics

How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book ... More

## Catalan Numbers with Applications

*Thomas Koshy*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

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 places, ... More

## General Theorems From Specific Images

*Marcus Giaquinto*

### in Visual Thinking in Mathematics

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter shows that some general arithmetical theorems almost certainly can be discovered by visualizing. First, it describes two examples of what look like acceptable routes to general theorems ... More

## Binomial Coefficients

*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/9780195334548.003.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

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 properties. Catalan ... More