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Schröder–Hipparchus Number
In combinatorics, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, and the number of ways of dissecting a convex polygon into smaller polygons by inserting diagonals. These numbers begin :1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, ... . They are also called the super-Catalan numbers, the little Schröder numbers, or the Hipparchus numbers, after Eugène Charles Catalan and his Catalan numbers, Ernst Schröder and the closely related Schröder numbers, and the ancient Greek mathematician Hipparchus who appears from evidence in Plutarch to have known of these numbers. Combinatorial enumeration applications The Schröder–Hipparchus numbers may be used to count several closely related combinatorial objects:.. *The ''n''th number in the sequence counts the number of different ways of subdividing of a polygon with ''n'' +&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Schröder (mathematician)
Friedrich Wilhelm Karl Ernst Schröder (25 November 1841 in Mannheim, Baden, Germany – 16 June 1902 in Karlsruhe, Germany) was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic, by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce. He is best known for his monumental ''Vorlesungen über die Algebra der Logik'' (''Lectures on the Algebra of Logic'', 1890–1905), in three volumes, which prepared the way for the emergence of mathematical logic as a separate discipline in the twentieth century by systematizing the various systems of formal logic of the day. Life Schröder learned mathematics at Heidelberg, Königsberg, and Zürich, under Otto Hesse, Gustav Kirchhoff, and Franz Neumann. After teaching school for a few years, he moved to the Technische Hochschule Darmstadt in 1874. Two years later, he took up a chair in math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edinburgh Mathematical Notes
''Edinburgh Mathematical Notes'' was a mathematics journal published from 1909 until 1961 by the Edinburgh Mathematical Society. The journal was originally named ''Mathematical Notes'', with the subtitle ''A Review of Elementary Mathematics and Science''.Edinburgh Mathematical Notes MacTutor History of Mathematics archive. Retrieved 22 October 2013. Its creation was the suggestion of George Alexander Gibson, a professor at , who wished to remove the more elementary or pedagogical articles from the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Narayana Number
In combinatorics, the Narayana numbers \operatorname(n, k), n \in \mathbb^+, 1 \le k \le n form a triangular array of natural numbers, called the Narayana triangle, that occur in various counting problems. They are named after Canadian mathematician T. V. Narayana (1930–1987). Formula The Narayana numbers can be expressed in terms of binomial coefficients: : \operatorname(n, k) = \frac Numerical values The first eight rows of the Narayana triangle read: Combinatorial interpretations Dyck words An example of a counting problem whose solution can be given in terms of the Narayana numbers \operatorname(n, k), is the number of words containing pairs of parentheses, which are correctly matched (known as Dyck words) and which contain distinct nestings. For instance, \operatorname(4, 2) = 6, since with four pairs of parentheses, six sequences can be created which each contain two occurrences the sub-pattern : (()(())) ((()())) ((())()) ()((())) (())(()) ((()) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The very first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.87. Notable publications * The 1972 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Row Vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, consisting of a single row of n entries, \boldsymbol a = \begin a_1 & a_2 & \dots & a_n \end. (Throughout this article, boldface is used for both row and column vectors.) The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: \begin x_1 \; x_2 \; \dots \; x_m \end^ = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end and \begin x_1 \\ x_2 \\ \vdots \\ x_m \end^ = \begin x_1 \; x_2 \; \dots \; x_m \end. The set of all row vectors with ''n'' entries in a given field (such as the real numbers) forms an ''n''-dimensional vector space; similarly, the set of all column vectors with ''m'' entries forms an ''m''-dimensional vector space. The space of row vectors with ''n'' entries can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronic Journal Of Combinatorics
The ''Electronic Journal of Combinatorics'' is a peer-reviewed open access scientific journal covering research in combinatorial mathematics. The journal was established in 1994 by Herbert Wilf (University of Pennsylvania) and Neil Calkin (Georgia Institute of Technology). The Electronic Journal of Combinatorics is a founding member of the Free Journal Network. According to the ''Journal Citation Reports'', the journal had a 2017 impact factor of 0.762. Editors-in-chief Current The current editors-in-chief are: * Maria Axenovich, Karlsruhe Institute of Technology, Germany * Miklós Bóna, University of Florida, United States * Julia Böttcher, London School of Economics, United Kingdom * Richard A. Brualdi, University of Wisconsin, Madison, United States * Eric Fusy, CNRS/LIX, École Polytechnique, France * Catherine Greenhill, UNSW Sydney, Australia * Brendan McKay, Australian National University, Australia * Bojan Mohar, Simon Fraser University, Canada * Marc Noy, Universitat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Pattern
In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation as a sequence of digits representing the result of applying the permutation to the digit sequence 123...; for instance the digit sequence 213 represents the permutation on three elements that swaps elements 1 and 2. If π and σ are two permutations represented in this way (these variable names are standard for permutations and are unrelated to the number pi), then π is said to ''contain'' σ as a ''pattern'' if some subsequence of the digits of π has the same relative order as all of the digits of σ. For instance, permutation π contains the pattern 213 whenever π has three digits ''x'', ''y'', and ''z'' that appear within π in the order ''x''...''y''...''z'' but whose values are ordered as ''y'' < ''x'' < ''z'', the same as the ordering of the values in the permutation 213. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Permutation
A double is a look-alike or doppelgänger; one person or being that resembles another. Double, The Double or Dubble may also refer to: Film and television * Double (filmmaking), someone who substitutes for the credited actor of a character * ''The Double'' (1934 film), a German crime comedy film * ''The Double'' (1971 film), an Italian film * ''The Double'' (2011 film), a spy thriller film * ''The Double'' (2013 film), a film based on the Dostoevsky novella * '' Kamen Rider Double'', a 2009–10 Japanese television series ** Kamen Rider Double (character), the protagonist in a Japanese television series of the same name Food and drink * Doppio, a double shot of espresso * Dubbel, a strong Belgian Trappist beer or, more generally, a strong brown ale * A drink order of two shots of hard liquor in one glass * A "double decker", a hamburger with two patties in a single bun Games * Double, action in games whereby a competitor raises the stakes ** , in contract bridge ** Doubl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijective Proof
In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective function that maps one set one-to-one onto the other. This technique can be useful as a way of finding a formula for the number of elements of certain sets, by corresponding them with other sets that are easier to count. Additionally, the nature of the bijection itself often provides powerful insights into each or both of the sets. Basic examples Proving the symmetry of the binomial coefficients The symmetry of the binomial coefficients states that : = . This means that there are exactly as many combinations of things in a set of size as there are combinations of things in a set of size . A bijective proof The key idea of the proof may be understood from a simple example: selecting children to be rewarded with ice cream cones, out of a group of children, has exactly th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge of has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of . The definition of the dual depends on the choice of embedding of the graph , so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
