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The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The n-th Catalan number can be expressed directly in terms of the central binomial coefficients by
Whilst the above is a concrete example Catalan numbers, similar problems can be evaluated using Fuss-Catalan formula: Computer Stack: ways of arranging and completing a computer stack of instructions, each time step 1 instruction is processed and p new instructions arrive randomly. If at the beginning of the sequence there are r instructions ...
Lobb numbers form a natural generalization of the Catalan numbers, which count the complete strings of balanced parentheses of a given length. Thus, the nth Catalan number equals the Lobb number L 0,n. [2] They are named after Andrew Lobb, who used them to give a simple inductive proof of the formula for the n th Catalan number. [3]
An important conjecture due to Catalan, sometimes called the Catalan–Dickson conjecture, is that every aliquot sequence ends in one of the above ways: with a prime number, a perfect number, or a set of amicable or sociable numbers. [3] The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats.
Substituting k = 1 into this formula gives the Catalan numbers and substituting k = 2 into this formula gives the Schröder–Hipparchus numbers. [7] In connection with the property of Schröder–Hipparchus numbers of counting faces of an associahedron, the number of vertices of the associahedron is given by the Catalan numbers.
The number of possible parenthesizations is given by the (n–1) th Catalan number, which is O(4 n / n 3/2), so checking each possible parenthesization (brute force) would require a run-time that is exponential in the number of matrices, which is very slow and impractical for large n. A quicker solution to this problem can be achieved by ...
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This number is given by the 5th Catalan number. It is trivial to triangulate any convex polygon in linear time into a fan triangulation, by adding diagonals from one vertex to all other non-nearest neighbor vertices. The total number of ways to triangulate a convex n-gon by non-intersecting diagonals is the (n−2)nd Catalan number, which equals