In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems; which often have a recursive flavour. They are named for the Belgian mathematician Eugène Charles Catalan (1814–1894). The nth Catalan number is given directly in terms of binomial coefficients by Catalan number - Properties of the Catalan numbers. One can verify that an alternative expression for Cn is ...

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• Catalan number - Properties of the Catalan numbers
• Catalan number - Applications in combinatorics
• Catalan number - Proof of the formula
• Catalan number - First proof: using generating functions
• Catalan number - Second proof
• Catalan number - Third proof
• Catalan number - Hankel matrix
• Catalan number - History

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List of geometric shapes - Not composed of circular arcs. Archimedean spiral astroid, paracycle, cubocycloid deltoid ellipse smoothed octagon super ellipse tomahawk ...

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List of geometric shapes, List of geometric shapes - Generally composed of straight line segments, List of geometric shapes - Curved, List of geometric shapes - Not composed of circular arcs

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By analogy, the subset of the Cartesian product X×X of any set X with itself, consisting of all pairs (x,x), is called the diagonal. It is the graph of the identity relation. It plays an important part in geometry: for example the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. Quite a major role is played in geometric studies by the idea of intersecting the diagonal with itself: not directly, but by perturbing it wi ...

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Diagonal, Diagonal - Polygons, Diagonal - Matrices, Diagonal - Geometry

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A monotone polygon is one with a boundary that consists of two parts, each of which consists of points that have incrementing coordinates in one dimension. Such a polygon can easily be triangulated in linear time as described by A. Fournier and D.Y. Montuno. To break up a polygon into monotone polygons, follow these steps: For each point, check if the vertices are both on the same side of the 'sweep line', a horizontal or vertical line. If they are, check the next sweep line on the other side. Break the polygon on the l ...

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Polygon triangulation, Polygon triangulation - Substracting ears method, Polygon triangulation - Using monotone polygons, Polygon triangulation - Reference

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There are several ways of explaining why the formula given for Cn is correct; that is, why it solves the combinatorial problems listed above. The first proof below uses a generating function, and is not particularly illuminating. The second and third proofs are examples of bijective proofs; they involve literally counting a collection of some kind of object to arrive at the correct formula. Catalan number - First proof: using generating functions. The Catalan numbe ...

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Catalan number, Catalan number - Properties of the Catalan numbers, Catalan number - Applications in combinatorics, Catalan number - Proof of the formula, Catalan number - First proof: using generating functions, Catalan number - Second proof, Catalan number - Third proof, Catalan number - Hankel matrix, Catalan number - History

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One way to triangulate a simple polygon is by using the assertion that any simple polygon without holes has at least two so called 'ears'. An ear is a triangle with two sides on the edge of the polygon and the other one completely inside it. The algorithm then consists of finding such an ear, removing it from the polygon (which results in a new polygon that still meets the conditions) and repeating until there is only one triangle left. This algorithm is pretty easy to implement, but imposes restrictions o ...

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Polygon triangulation, Polygon triangulation - Substracting ears method, Polygon triangulation - Using monotone polygons, Polygon triangulation - Reference

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As applied to a polygon, a diagonal is a line segment joining two vertices that are not adjacent. Therefore a quadrilateral has two diagonals, joining opposite pairs of vertices. For a convex polygon the diagonals run inside the polygon. This is not so for re-entrant polygons. In fact a polygon is convex if and only if the diagonals are internal. When n is the number of vertices in a polygon and d is the number of possible different diagonals, each vertex has possible diagonals to all other vertices save for itsel ...

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Diagonal, Diagonal - Polygons, Diagonal - Matrices, Diagonal - Geometry

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One can verify that an alternative expression for Cn is This shows that Cn is a natural number, which is not a priori obvious from the first formula given. This expression forms the basis for André's proof of the correctness of the formula (see below under second proof). The first Catalan numbers (sequence A000108 in OEIS) for n = 0, 1, 2, 3, ... are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 2446626702 ...

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Catalan number, Catalan number - Properties of the Catalan numbers, Catalan number - Applications in combinatorics, Catalan number - Proof of the formula, Catalan number - First proof: using generating functions, Catalan number - Second proof, Catalan number - Third proof, Catalan number - Hankel matrix, Catalan number - History

Read more here: » Catalan number: Encyclopedia II - Catalan number - Properties of the Catalan numbers