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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
The Fuss-Catalan represents the number of legal permutations or allowed ways of arranging a number of articles, that is restricted in some way. This means that they are related to the Binomial Coefficient. The key difference between Fuss-Catalan and the Binomial Coefficient is that there are no "illegal" arrangement permutations within Binomial ...
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.
Generating functions were first introduced by Abraham de Moivre in 1730, ... th Catalan number. Introducing a free parameter (snake oil method) Sometimes ...
In combinatorial mathematics, the Lobb number L m,n counts the ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses. [1] Lobb numbers form a natural generalization of the Catalan numbers, which count the complete strings of balanced parentheses of a given ...
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
As eye-catching as the 2025 slate of films from consecrated Catalan filmmakers is, the number of titles that will come from first-time feature directors is equally impressive.
The rule in Catalan is to follow the number with the last letter in the singular and the last two letters in the plural. [24] Most numbers follow the pattern exemplified by vint '20' (20è m sg, 20a f sg, 20ns m pl, 20es f pl), but the first few ordinals are irregular, affecting the abbreviations of the masculine forms. Superscripting is not ...