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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] |
Catalan Number 4x4 Grid Example
Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid #13178, named "Catalan" * Catalán (crater), a lunar crater named for Miguel Ángel Catalán * Çatalan, İvrindi, a village in Balıkesir province, Turkey * Çatalan, Karaisalı, a village in Adana Province, Turkey * Catalan Bay, Gibraltar * Catalan Sea, more commonly known as the Balearic Sea * Catalan Mediterranean System, the Catalan Mountains Facilities and structures * Çatalan Bridge, Adana, Turkey * Çatalan Dam, Adana, Turkey * Catalan Batteries, Gibraltar People * Catalan, Lord of Monaco (1415–1457), Lord of Monaco from 1454 until 1457 * Alfredo Catalán (born 1968), Venezuelan politician * Alex Catalán (born 1968), Spanish filmmaker * Arnaut Catalan (1219–1253), troubador * Diego Catalán (1928–2008), Spanish philologist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Path
In combinatorics, a lattice path in the -dimensional integer lattice of length with steps in the set , is a sequence of vectors such that each consecutive difference v_i - v_ lies in . A lattice path may lie in any lattice in , but the integer lattice is most commonly used. An example of a lattice path in of length 5 with steps in S = \lbrace (2,0), (1,1), (0,-1) \rbrace is L = \lbrace (-1,-2), (0,-1), (2,-1), (2,-2), (2,-3), (4,-3) \rbrace . North-East lattice paths A North-East (NE) lattice path is a lattice path in \mathbb^2 with steps in S = \lbrace (0,1), (1,0) \rbrace . The (0,1) steps are called North steps and denoted by N 's; the (1,0) steps are called East steps and denoted by E 's. NE lattice paths most commonly begin at the origin. This convention allows us to encode all the information about a NE lattice path L in a single permutation word. The length of the word gives us the number of steps of the lattice path, k . The order of the N 's an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a Disjoint union of graphs, disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan 4 Leaves Binary Tree Example
Catalan may refer to: Catalonia From, or related to Catalonia: * Catalan language, a Romance language * Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia Places * 13178 Catalan, asteroid #13178, named "Catalan" * Catalán (crater), a lunar crater named for Miguel Ángel Catalán * Çatalan, İvrindi, a village in Balıkesir province, Turkey * Çatalan, Karaisalı, a village in Adana Province, Turkey * Catalan Bay, Gibraltar * Catalan Sea, more commonly known as the Balearic Sea * Catalan Mediterranean System, the Catalan Mountains Facilities and structures * Çatalan Bridge, Adana, Turkey * Çatalan Dam, Adana, Turkey * Catalan Batteries, Gibraltar People * Catalan, Lord of Monaco (1415–1457), Lord of Monaco from 1454 until 1457 * Alfredo Catalán (born 1968), Venezuelan politician * Alex Catalán (born 1968), Spanish filmmaker * Arnaut Catalan (1219–1253), troubador * Diego Catalán (1928–2008), Spanish philologist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a Disjoint union of graphs, disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tree
In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple (''L'', ''S'', ''R''), where ''L'' and ''R'' are binary trees or the empty set and ''S'' is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary (and K-ary) trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence—a term which appears in some very old programming books, before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of ''binary tree'' to emphasize the fact that the tree is rooted, bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tamari Lattice, Trees
Tamari may refer to: * A type of soy sauce, produced mainly in the Chūbu region of Japan * Dov Tamari (mathematician) ** Tamari lattice * Dov Tamari (brigadier general) Dov Tamari ( he, דב תמרי; born 1936) was an Israeli military leader. He retired as a brigadier general. He held several prominent posts, including the IDF's Chief Intelligence Officer. Military service Tamari was drafted into the IDF in ... See also * Tammari (other) * Temari (other) {{disambiguation, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Chain Multiplication
Matrix chain multiplication (or the matrix chain ordering problem) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to ''perform'' the multiplications, but merely to decide the sequence of the matrix multiplications involved. The problem may be solved using dynamic programming Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. .... There are many options because matrix multiplication is associative. In other words, no matter how the product is parenthesized, the result obtained will remain the same. For example, for four matrices ''A'', ''B'', ''C'', and ''D'', there are five possible options: :((''AB'')''C'')''D'' = (''A''(''BC''))''D'' = (''AB'')(''CD'') = ''A''((''BC'')''D'') = ''A''(''B''(''CD'')) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Operator
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation ''on a set'' is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures that are studied ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
