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Catalan's Problem
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyck Word
In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of square brackets and The set of Dyck words forms the Dyck language. Dyck words and language are named after the mathematician Walther von Dyck. They have applications in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions. Formal definition Let \Sigma = \ be the alphabet consisting of the symbols and Let \Sigma^ denote its Kleene closure. The Dyck language is defined as: : \. Context-free grammar It may be helpful to define the Dyck language via a context-free grammar in some situations. The Dyck language is generated by the context-free grammar with a single non-terminal , and the production: : That is, ''S'' is either the empty string () or is " , an element of the Dyck language, the matching ", and an element of the Dyck language. An alternative context-free grammar for the D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
