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Topic: Catalan number

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	Catalan numbers - Wikipedia, the free encyclopedia
		Catalan number and the expression on the right tends towards 1 for n → ∞.
		is the number of stack-sortable permutations of {1,..., n}.
		The Catalan sequence was first described in the 18th century by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles.
	en.wikipedia.org /wiki/Catalan_number (1686 words)

	What's Special About This Number?
		is the number of planar partitions of 10.
		is the number of planar partitions of 11.
		is the number of planar partitions of 12.
	www.stetson.edu /~efriedma/numbers.html (7399 words)

	Catalan number: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-07)
		In mathematics, in particular in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural...
		1 (one) is a number, numeral, and glyph....
		The bell numbers, named in honor of eric temple bell, are the number of distinct possible ways of putting n labelled balls in to one or more anonymous boxes....
	www.absoluteastronomy.com /encyclopedia/c/ca/catalan_number.htm (2057 words)

	Catalan - Wikipedia, the free encyclopedia
		A Catalan speaker, whether or not from Catalonia proper (see Catalan Countries).
		Catalan forge, an early type of open-hearth furnace
		Catalan vault, an architectural feature (also known as a Catalan arch or a Catalan turn)
	en.wikipedia.org /wiki/Catalan (145 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-07)
		Date: 12/16/1999 at 10:36:52 From: Doctor Anthony Subject: Re: Catalan Numbers A physical interpretation is to consider an n x n square grid of streets and to find the number of routes from one corner of the square to the corner diagonally opposite that do not cross the diagonal.
		Catalan Numbers --------------- The number of sequences a(1), a(2), a(3),..., a(2n) of 2n terms that can be formed by using n +1's and n -1's whose partial sums satisfy a(1) + a(2) +...
		The number of sequences of (n+1) +1's and (n-1) -1's is the number (2n)!/[(n+1)!(n-1)!] = C(2n,n-1).
	mathforum.org /dr.math/problems/bradley.12.15.99.html (824 words)

	Catalan number
		The Catalan numbers form a sequence of natural numbers that occur in various counting problems in combinatorics.
		is also equal to the number of different ways a polygon with n+2 sides can be cut into triangles by connecting vertices with straight lines.
		The number of stack-sortable permutations of {1,...,n} is equal to C
	www.ebroadcast.com.au /lookup/encyclopedia/dy/Dyck_word.html (607 words)

	PlanetMath: Catalan numbers (Site not responding. Last check: 2007-11-07)
		The ordinary generating function for the Catalan numbers is
		The Catalan sequence is named for Eugène Charles Catalan, but it was discovered in 1751 by Euler when he was trying to solve the problem of subdividing polygons into triangles.
		This is version 5 of Catalan numbers, born on 2002-02-27, modified 2004-06-28.
	planetmath.org /encyclopedia/CatalanSequence.html (134 words)

	1000000 (number) - Wikipedia, the free encyclopedia
		The million is sometimes used in the English language as a metaphor for a very large number, as in "Never in a million years" and "You're one in a million", or a hyperbole, as in "I've walked a million miles".
		The word "million" is common to the short scale and long scale numbering systems (and also to the proposed Rowlett numbering system), unlike the larger numbers, which have different names in the two systems.
		1048576 = 2^20 (power of two), 2116-gonal number, an 8740-gonal number and a 174764-gonal number, the number of bytes in a mebibyte, the number of kibibytes in a gibibyte, and so on.
	en.wikipedia.org /wiki/Million (477 words)

	Catalan numbers - (Site not responding. Last check: 2007-11-07)
		Catalan number and the expression on the right tends towards 1 for n → ∞.
		is the number of different ways n + 1 factors can be completely parenthesized (or the number of ways of associating n applications of a binary operator).
		The n×n Hankel matrix whose (i, j) entry is the Catalan number C
	psychcentral.com /psypsych/Catalan_number (1781 words)

	The Catalan Numbers (Site not responding. Last check: 2007-11-07)
		The Catalan Numbers are a sequence of numbers which show up in many contexts.
		In 1838, Eugene Catalan, a Belgian mathematician, solved the problem of how many ways one can parenthesize a chain of N+1 letters using N pairs of parentheses such that there are either two letters, a parenthesized expression and a letter, or two parenthesized expressions within each pair of parentheses.
		The final method of generating Catalan Numbers that I examine involves the movement of a rook.
	www.saintanns.k12.ny.us /depart/math/Seth/intro.html (378 words)

	[No title]
		The Catalan series is the series of derived from the factors of the central numbers from Pascal's Triangle.
		This is the series of numbers which are equal to the sum of all of its positive divisors, excluding itself.
		All the variations for that number are 5678, 5687, 5768, 5786, 5867, 5885, 6578, 6587, 6758, 6785, 6857, 6875, 7568, 7586, 7658, 7685, 7867, 7876, 8567, 8576, 8657, 8675, 8756 and 8765.
	members.lycos.co.uk /brisray/qbasic/qnumber.htm (2974 words)

	Cosmology ... pelastrated numbers ... new math?
		In each hexagon is mentioned the number of combinations (number of sub-sets with a unique layer combination) that are possible with the previous sub-sets.
		When we analyze this in the previous sheet on the number 5 we find that there are 4 combinations of 3 with 2, and 10 combinations of 1 with 4, and that confirms the 4 and 10 found in the Pascal Triangle.
		When we bring each catalan number to power 2 we get all pelastration combinations possible by those unique holons, on that level of generation, and this gives us all new unique combination of the next generation on which we much add the combinations of all previous generations.
	www.mu6.com /numbers.html (1128 words)

	Relation Pascal Triangle and Catalan Numbers
		Eugène Charles Catalan (born: 30 May 1814 in Bruges, Belgium - died: 14 Feb 1894 in Liège, Belgium) defined the numbers, called today the Catalan numbers, while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals.
		Catalan was not the first to solve the problem, however, since Segner had solved it in the 18 th century, although his solution was not as elegant as Catalan's.
		They were discovered by Leonhard Euler when he was attempting to find a general formula to express the number of ways to divide a polygon with N sides into triangles using non-intersecting diagonals.
	milan.milanovic.org /math/english/fibo/fibo4.html (287 words)

	Problem 43 . Catalan Numbers
		At k=77605, m=41298 there is a number which survives my initial tests against a few small primes but will inevitably fall to a wider test.
		The nth Catalan number, from the formula, is a factor of (2n)!.
		For n>6 the answer is always "Yes" and as a result the Catalan numbers are never prime, nor are they a single odd prime times a power of 2.
	www.primepuzzles.net /problems/prob_043.htm (1015 words)

	Cosmology ... and Catalan numbers. (Site not responding. Last check: 2007-11-07)
		You can see in next Pascal Triangle that each Catalan number is the sum of specific Pascal numbers.
		The Catalan numbers occupy all Pascal numbers except the middle row and the rows of ones.
		I mean the Catalan number 58,786 (a tribe) is composed by 13 families (red hexgons) of sub-sets.
	www.mu6.com /numbers_catalan.html (369 words)

	Robert M. Dickau - Catalan Numbers (Site not responding. Last check: 2007-11-07)
		Eugène Charles Catalan (1814-1894), arise in a number of problems in combinatorics.
		the number of ways in which parentheses can be placed in a sequence of numbers to be multiplied, two at a time;
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /advanced/robertd/catalan.html (173 words)

	TopCoder Training Camp
		C(N) := The number of possible triangulations (with N triangles) of a convex polygon with N+2 sides.
		Again, it should be fairly easy to see that the number of mountains is equal to the number of "correct" terms of parentheses (for example, "())(" is not correct, because the second closing parenthesis is not closing any preceding opening parenthesis).
		Btw, the standard text book Introduction to Algorithms also covers the Catalan Numbers in a nice problem (well, the problem uses only formal math, no visualization, so it's not the nicest problem I've ever seen).
	www.stefan-pochmann.de /spots/tutorials/catalan_numbers (1131 words)

	COMPUTATIONAL GEOMETRY PROJECT- ENUMERATION OF ALL POSSIBLE UNQIUE TRIANGULATIONS OF A CONVEX POLYGON
		Catalan (N) = (1/N) * Choose (2N,N) The (N-2)th Catalan Number, which is the number of triangulations of an N-gon is:
		The number of sons of t is dependent on the outdegree of the vertex Vn.
		Thereafter we add in all edges E(p,n) of the parent, where p was a vertex numbered higher than 'i' (remember we are constructing the 'i'th son).
	cgm.cs.mcgill.ca /~athens/cs507/Projects/2002/AjitRajwade (2602 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		function c = catalan (n) %% CATALAN computes the Catalan numbers, from C(0) to C(N).
		% % Parameters: % % Input, integer N, the number of Catalan numbers desired.
		% % Output, integer C(1:N+1), the Catalan numbers from C(0) to C(N).
	www.csit.fsu.edu /~burkardt/m_src/subset/catalan.m (217 words)

	Numbers
		Two numbers n and m are called an amicable pair if the sum of all positive divisors of n is equal to the sum of all positive divisors of m and both are equal to n + m.
		Pentagonal numbers to pentagons is the same as triangular numbers to triangles and square numbers to squares.
		Definition: The number n is called a weird number if it is abundant, but it is not the sum of any subset of its proper factors.
	www.tanyakhovanova.com /Numbers/numbers.html (1913 words)

	Math Forum: Pascal's Triangle
		The Catalan Numbers express the number of ways you can divide a polygon with N sides into triangles, using non-intersecting diagonals.
		These numbers are today called the Catalan numbers after the Belgian Eugène Charles Catalan, who however was not the first to solve the problem, since
		Seth Johnson writes that in 1838, Catalan "solved the problem of how many ways one can parenthesize a chain of N+1 letters using N pairs of parentheses such that there are either two letters, a parenthesized expression and a letter, or two parenthesized expressions within each pair of parentheses.
	mathforum.org /workshops/usi/pascal/pascal_catalan.html (210 words)

	The Math Forum - Math Library - Fibonacci Sequence (Site not responding. Last check: 2007-11-07)
		Fibonacci numbers are closely related to the golden ratio (also known as the golden mean, golden number, golden section) and golden string.
		Catalan number diagrams; permutation diagrams; derangements; shortest-path diagrams; Stirling numbers of the first and second kind; Bell numbers, harmonic numbers and the book-stacking...more>>
		A new mathematical constant, from a branch of the Fibonacci number family in which, instead of always adding two terms to produce the next term, you either add or subtract, depending on the flip of a coin at each stage in the calculation.
	www.mathforum.org /library/topics/fibonacci (1711 words)

	cs2223 Exam 3 (Site not responding. Last check: 2007-11-07)
		The Catalan number is defined in the text - page 280 - as:
		Write a complete, working, non-recursive, C or C++ function which calculates the Catalan number.
		The Catalan numbers grow very quickly so, to fit the answer in a
	www.cs.wpi.edu /~cs2223/d98/exams/exam03/exam3.html (215 words)

	Catalan Workshop: Menorca Sep 1-5, 2004
		Under the auspices of the GTEM Galois Theory and Explicit Methods in Arithmetic European project.
		The workshop will focus on a proof of the Catalan conjecture.
		The conjecture was made by Belgian mathematician Eugénie Charles Catalan
	abel.math.harvard.edu /~lario/Catalan.html (183 words)

	catalan number (Site not responding. Last check: 2007-11-07)
		Offers catalan number with a product or service that delivers catalan dictionary english and catalan espanol online traductor.
		Language Cds and language tapes to help you gain language immersion Offers catalan number with a product or service that delivers catalan dictionary english and catalan espanol online traductor.
		Advanced language instruction with electronic dictionaries and translation software Offers catalan number with a product or service that delivers catalan dictionary english and catalan espanol online traductor.
	www.languageresourceonline.com /Catalan/catalan-number.html (140 words)

	Four holons of the fifth catalan number level
		Four holons of the fifth catalan number level
		As the Catalan Numbers show us there are on the fifth level 14 new combinations possible.
		Next page we see the two last holon types 5M and 5N of catalan number level 5.
	www.mu6.com /holons_5i_to_5L.html (124 words)

	Citebase - What power of two divides a weighted Catalan number?
		is the height of the ith ascent of P. The corresponding weighted Catalan number is C
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
		Congruences for Catalan and Motzkin numbers and related sequences.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0601339 (315 words)

	cs2223 Exam 3 Solution (Site not responding. Last check: 2007-11-07)
		This is Newton's method for finding the square root of a number.
		The recurrence relation for the number of additions or subtractions is:
		The attached script shows that this function produces the correct answer when n=20.
	web.cs.wpi.edu /~cs2223/d98/exams/exam03/solution3.html (182 words)

	Confirming Kleitman-Winston Conjecture ON THE LARGEST COEFFICIENT IN A q-CATALAN NUMBER (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies.
		Abstract: In 1983 Kleitman and Winston conjectured that the largest coefficient in an n-th q-Catalan number is of order O(4 n /n 3/2).
		Assuming its truth, they proved that the total number of n-tournament score sequences is O(4 n /n 5/2), thus matching their own lower bound.
	citeseer.ist.psu.edu /348400.html (310 words)

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