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The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers.
Lattice of the 14 Dyck words of length 8 - and interpreted as up and down Cn is the number of Dyck words  of length 2n.
A Dyck word is a string consisting of n X's and n Y's such that no initial segment of the string has more Y's than X's. For example, the following are the Dyck words of length 6: Re-interpreting the symbol X as an open parenthesis and Y as a close parenthesis, Cn counts the number of expressions containing n pairs of parentheses which are correctly matched: A rooted binary tree is full if every vertex has either two children or no children.
An ordered tree is a rooted tree in which the children of each vertex are given a fixed left-to-right order. A monotonic path is one which starts in the lower left corner, finishes in the upper right corner, and consists entirely of edges pointing rightwards or upwards.
Counting such paths is equivalent to counting Dyck words: X stands for "move right" and Y stands for "move up".
This can be succinctly represented by listing the Catalan elements by column height: The number of triangles formed is n and the number of different ways that this can be achieved is Cn.
A fortioriCn never exceeds the nth Bell number. This law is important in free probability theory and the theory of random matrices. Cn is the number of ways to tile a stairstep shape of height n with n rectangles. Cn is the number of ways to form a "mountain range" with n upstrokes and n downstrokes that all stay above a horizontal line.
The mountain range interpretation is that the mountains will never go below the horizon. Cn is the number of standard Young tableaux whose diagram is a 2-by-n rectangle.
As such, the formula can be derived as a special case of the hook-length formula. Cn is the number of ways that the vertices of a convex 2n-gon can be paired so that the line segments joining paired vertices do not intersect. This is precisely the condition that guarantees that the paired edges can be identified sewn together to form a closed surface of genus zero a topological 2-sphere.
Cn is the number of semiorders on n unlabeled items.Mathematics Mathematical concepts named after mathematician Eugène Catalan: Catalan numbers, a sequence of natural numbers that occur in .
Various Number Theorists' Home Pages/Departmental listings Complete listing [ A | B | C | D | E | F | G | H | I | J | K | L | M] [ N | O | P | Q | R | S | T | U | V.
Empedocles of Acragas (c. BC) Inventor of rhetoric and borderline charlatan. His arbitrary explanation of reality with 4 elements (Earth, Air, Fire and Water) and 2 forces (Love and Strife) dominated Western thought for over two millenia.
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Understanding Catalan’s Conjecture. Posted on October 9, by mathematicians come up with profound ideas and formulations that impact many fields within mathematics.
Let’s go ahead and peel this metaphorical onion, shall we? What is the conjecture?
This marvelous conjecture was proposed by the Belgian mathematician, . Dear Twitpic Community - thank you for all the wonderful photos you have taken over the years.
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