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The number of distinct Dyck words with exactly n pairs of parentheses is the n-th Catalan number. Notice that the Dyck language of words with n parentheses pairs is equal to the union, over all possible k, of the Dyck languages of words of n parentheses pairs with k innermost pairs, as defined in
The proof that the language of balanced (i.e., properly nested) parentheses is not regular follows the same idea. Given p {\displaystyle p} , there is a string of balanced parentheses that begins with more than p {\displaystyle p} left parentheses, so that y {\displaystyle y} will consist entirely of left parentheses.
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 ...
The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced sets of parentheses). The free monoid on A {\displaystyle A} (where | A | > 1 {\displaystyle \left|A\right|>1} ) is the syntactic monoid of the language { w w R ∣ w ∈ A ∗ } {\displaystyle \{ww^{R}\mid w\in A^{*}\}} , where w R {\displaystyle w^{R ...
In theoretical computer science and formal language theory, a regular language (also called a rational language) [1] [2] is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expression engines, which are augmented with features that allow the recognition of non-regular languages).
The sequence of pushes and pops performed by Knuth's sorting algorithm as it sorts a stack-sortable permutation form a Dyck language: reinterpreting a push as a left parenthesis and a pop as a right parenthesis produces a string of balanced parentheses. Moreover, every Dyck string comes from a stack-sortable permutation in this way, and every ...
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A string is a substring (or factor) [1] of a string if there exists two strings and such that =.In particular, the empty string is a substring of every string. Example: The string = ana is equal to substrings (and subsequences) of = banana at two different offsets: