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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
In some publications this equation is sometimes referred to as Two-parameter Fuss–Catalan numbers or Raney numbers. The implication is the single-parameter Fuss-Catalan numbers are when r = 1 {\displaystyle \,r=1\,} and p = 2 {\displaystyle \,p=2\,} .
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]
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.
A page from Minggatu's Geyuan Milü Jiefa Minggatu's geometrical model for trigonometric infinite series Minggatu discovered Catalan numbers. Minggatu (Mongolian script: ᠮᠢᠩᠭᠠᠲᠦ; Chinese: 明安图; pinyin: Míng'āntú, c.1692-c. 1763), full name Sharavyn Myangat (Mongolian: Шаравын Мянгат), also known as Ming Antu, was a Mongolian astronomer, mathematician, and ...
The smallest integer m > 1 such that p n # + m is a prime number, where the primorial p n # is the product of the first n prime numbers. A005235 Semiperfect numbers
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
A case can be made that Percy MacMahon () was the first to prove a result in the field with his study of "lattice permutations". [1] In particular MacMahon shows that the permutations which can be divided into two decreasing subsequences (i.e., the 123-avoiding permutations) are counted by the Catalan numbers.