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/9780195334548.003.0012
- 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 ...
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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 Jonah's theorem.Less
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 Jonah's theorem.
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, ...
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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, abundant applications, and intriguing properties, they have fascinated amateurs and mathematicians alike. However, Catalan numbers are even more fascinating. Like the North Star in the evening sky, they are a beautiful and bright light in the mathematical heavens. They continue to provide a fertile ground for number theorists, especially, Catalan enthusiasts and computer scientists. Since the publication of Euler's triangulation problem (1751) and Catalan's parenthesization problem (1838), over 400 articles and problems on Catalan numbers have appeared in various periodicals. As Martin Gardner noted, even though many amateurs and mathematicians may know the abc's of Catalan sequence, they may not be familiar with their myriad unexpected occurrences, delightful applications, properties, or the beautiful and surprising relationships among numerous examples. Like Fibonacci and Lucas numbers, Catalan numbers are also an excellent source of fun and excitement. They can be used to generate interesting dividends for students, such as intellectual curiosity, experimentation, pattern recognition, conjecturing, and problem-solving techniques. The central character in the nth Catalan number is the central binomial coefficient. So, Catalan numbers can be extracted from Pascal's triangle. In fact, there are a number of ways they can be read from Pascal's triangle; every one of them is described and exemplified. This brings Catalan numbers a step closer to number-theory enthusiasts, especially.Less
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, abundant applications, and intriguing properties, they have fascinated amateurs and mathematicians alike. However, Catalan numbers are even more fascinating. Like the North Star in the evening sky, they are a beautiful and bright light in the mathematical heavens. They continue to provide a fertile ground for number theorists, especially, Catalan enthusiasts and computer scientists. Since the publication of Euler's triangulation problem (1751) and Catalan's parenthesization problem (1838), over 400 articles and problems on Catalan numbers have appeared in various periodicals. As Martin Gardner noted, even though many amateurs and mathematicians may know the abc's of Catalan sequence, they may not be familiar with their myriad unexpected occurrences, delightful applications, properties, or the beautiful and surprising relationships among numerous examples. Like Fibonacci and Lucas numbers, Catalan numbers are also an excellent source of fun and excitement. They can be used to generate interesting dividends for students, such as intellectual curiosity, experimentation, pattern recognition, conjecturing, and problem-solving techniques. The central character in the nth Catalan number is the central binomial coefficient. So, Catalan numbers can be extracted from Pascal's triangle. In fact, there are a number of ways they can be read from Pascal's triangle; every one of them is described and exemplified. This brings Catalan numbers a step closer to number-theory enthusiasts, especially.
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/9780195334548.003.0001
- 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 ...
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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 numbers Cn are introduced.Less
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 numbers Cn are introduced.
Kurt Smith
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199583652
- eISBN:
- 9780191723155
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199583652.003.0013
- Subject:
- Philosophy, History of Philosophy, Metaphysics/Epistemology
This chapter shows how the combinatorial nature of bodies expresses the permutation group concept, the latter expressing the conditions underwriting a genuine mathematical system. The chapter ...
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This chapter shows how the combinatorial nature of bodies expresses the permutation group concept, the latter expressing the conditions underwriting a genuine mathematical system. The chapter explains how synthesis, or a synthetic system, is isomorphic to a permutation group. What is more, it is shown how analysis, or the system of concepts resulting from analysis, is isomorphic to a synthetic system. This, it is argued, establishes a sense in which analysis and synthesis are ‘flip sides’ of the same conceptual coin. Since an analytic‐synthetic system is a group, and a group is a genuine mathematical system, it is seen the important role that Descartes's enumeration played in establishing a ‘mathematized’ physics.Less
This chapter shows how the combinatorial nature of bodies expresses the permutation group concept, the latter expressing the conditions underwriting a genuine mathematical system. The chapter explains how synthesis, or a synthetic system, is isomorphic to a permutation group. What is more, it is shown how analysis, or the system of concepts resulting from analysis, is isomorphic to a synthetic system. This, it is argued, establishes a sense in which analysis and synthesis are ‘flip sides’ of the same conceptual coin. Since an analytic‐synthetic system is a group, and a group is a genuine mathematical system, it is seen the important role that Descartes's enumeration played in establishing a ‘mathematized’ physics.
Marcel Danesi
- Published in print:
- 2020
- Published Online:
- January 2020
- ISBN:
- 9780198852247
- eISBN:
- 9780191886959
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198852247.003.0005
- Subject:
- Mathematics, History of Mathematics, Educational Mathematics
Mathematicians have devised notations and symbols to compress information and to explore mathematics via the symbols themselves. In the 1500s the invention of exponential notation, which was devised ...
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Mathematicians have devised notations and symbols to compress information and to explore mathematics via the symbols themselves. In the 1500s the invention of exponential notation, which was devised as a type of shorthand to facilitate the cumbersomeness of just reading repeated multiplications of the same digit, was a watershed event. Exponential notation not only saves space and lessens the mental energy required to process the relevant information, it is critical for writing larger and larger numbers. Moreover, mathematicians started to play with exponential notation in an abstract way, discovering new facts about numbers, leading to the notion of logarithms and all the discoveries that this, in turn, has brought about. This chapter will deal with exponents and logarithms, and their significance to the history of mathematics—a history often characterized by problems of notation that have led serendipitously to new ideas and branches.Less
Mathematicians have devised notations and symbols to compress information and to explore mathematics via the symbols themselves. In the 1500s the invention of exponential notation, which was devised as a type of shorthand to facilitate the cumbersomeness of just reading repeated multiplications of the same digit, was a watershed event. Exponential notation not only saves space and lessens the mental energy required to process the relevant information, it is critical for writing larger and larger numbers. Moreover, mathematicians started to play with exponential notation in an abstract way, discovering new facts about numbers, leading to the notion of logarithms and all the discoveries that this, in turn, has brought about. This chapter will deal with exponents and logarithms, and their significance to the history of mathematics—a history often characterized by problems of notation that have led serendipitously to new ideas and branches.
