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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.
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.
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]
Eugène Charles Catalan (French pronunciation: [øʒɛn ʃaʁl katalɑ̃]; 30 May 1814 – 14 February 1894) [2] was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics.
The following other wikis use this file: Usage on cs.wikipedia.org Catalanova čísla; Usage on de.wikipedia.org Catalan-Zahl; Diskussion:Catalan-Zahl
The only known Catalan pseudoprimes are: 5907, 1194649, and 12327121 (sequence A163209 in the OEIS) with the latter two being squares of Wieferich primes. In general, if p is a Wieferich prime, then p 2 is a Catalan pseudoprime.
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
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