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Stirling Permutation
In combinatorial mathematics, a Stirling permutation of order ''k'' is a permutation of the multiset 1, 1, 2, 2, ..., ''k'', ''k'' (with two copies of each value from 1 to ''k'') with the additional property that, for each value ''i'' appearing in the permutation, the values between the two copies of ''i'' are larger than ''i''. For instance, the 15 Stirling permutations of order three are :1,1,2,2,3,3; 1,2,2,1,3,3; 2,2,1,1,3,3; :1,1,2,3,3,2; 1,2,2,3,3,1; 2,2,1,3,3,1; :1,1,3,3,2,2; 1,2,3,3,2,1; 2,2,3,3,1,1; :1,3,3,1,2,2; 1,3,3,2,2,1; 2,3,3,2,1,1; :3,3,1,1,2,2; 3,3,1,2,2,1; 3,3,2,2,1,1. The number of Stirling permutations of order ''k'' is given by the double factorial (2''k'' − 1)!!. Stirling permutations were introduced by in order to show that certain numbers (the numbers of Stirling permutations with a fixed number of descents) are non-negative. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements and , but vary in the multiplicities of their elements: * The set contains only elements and , each having multiplicity 1 when is seen as a multiset. * In the multiset , the element has multiplicity 2, and has multiplicity 1. * In the multiset , and both have multiplicity 3. These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, order does not matter in discriminating multisets, so and denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the double factorial is :n!! = \prod_^\frac (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,, and for odd it is :n!! = \prod_^\frac (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,. For example, . The zero double factorial as an empty product. The sequence of double factorials for even = starts as : 1, 2, 8, 48, 384, 3840, 46080, 645120,... The sequence of double factorials for odd = starts as : 1, 3, 15, 105, 945, 10395, 135135,... The term odd factorial is sometimes used for the double factorial of an odd number. History and usage In a 1902 paper, the physicist Arthur Schuster wrote: states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Number
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book ''Methodus differentialis'' (1730). They were rediscovered and given a combinatorial meaning by Masanobu Saka in 1782. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind. Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind. Each kind is detailed in its respective article, this one serving as a description of relations between them. A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of ''n'' elements into ''k'' non-empty subsets, with different ways of counting orderings within each subset. Notation Sev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Stirling (mathematician)
James Stirling (11 May O.S. 1692, Garden, Stirlingshire – 5 December 1770, Edinburgh) was a Scottish mathematician. He was nicknamed "The Venetian". The Stirling numbers, Stirling permutations, and Stirling's approximation are named after him. He also proved the correctness of Isaac Newton's classification of cubics. Biography Stirling was born on 11 May 1692 O.S. at Garden House near Stirling, the third son of Archibald Stirling, Lord Garden. At 18 years of age he went to Balliol College, Oxford, where, chiefly through the influence of the Earl of Mar, he was nominated in 1711 to be one of Bishop Warner's exhibitioners (or Snell exhibitioner) at Balliol. In 1715 he was expelled on account of his correspondence with his cousins, who were members of the Keir and Garden families, who were noted Jacobites, and had been accessory to the " Gathering of the Brig o' Turk" in 1708. From Oxford he made his way to Venice, where he occupied himself as a professor of mathematics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Permutation Euler Tour
Stirling (; sco, Stirlin; gd, Sruighlea ) is a city in central Scotland, northeast of Glasgow and north-west of Edinburgh. The market town, surrounded by rich farmland, grew up connecting the royal citadel, the medieval old town with its merchants and tradesmen, the Old Bridge and the port. Located on the River Forth, Stirling is the administrative centre for the Stirling council area, and is traditionally the county town of Stirlingshire. Proverbially it is the strategically important "Gateway to the Highlands". It has been said that "Stirling, like a huge brooch clasps Highlands and Lowlands together". Similarly "he who holds Stirling, holds Scotland" is often quoted. Stirling's key position as the lowest bridging point of the River Forth before it broadens towards the Firth of Forth made it a focal point for travel north or south. When Stirling was temporarily under Anglo-Saxon sway, according to a 9th-century legend, it was attacked by Danish invaders. The sound of a ... [...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] |
Euler Tour Technique
The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree. The ETT allows for efficient, parallel computation of solutions to common problems in algorithmic graph theory. It was introduced by Tarjan and Vishkin in 1984. Construction Given an undirected tree presented as a set of edges, the Euler tour representation (ETR) can be constructed in parallel as follows: * We construct a symmetric list of directed edges: ** For each undirected edge in the tree, insert (''u'',''v'') and (''v'',''u'') in the edge list. * Sort the edge list lexicographically. (Here we assume that the nodes of the tree are ordered, and that the root is the first element in this order.) * Construct adjacency lists for each nod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langford Pairing
In combinatorial mathematics, a Langford pairing, also called a Langford sequence, is a permutation of the sequence of 2''n'' numbers 1, 1, 2, 2, ..., ''n'', ''n'' in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of each number ''k'' are ''k'' units apart. Langford pairings are named after C. Dudley Langford, who posed the problem of constructing them in 1958. Langford's problem is the task of finding Langford pairings for a given value of ''n''. The closely related concept of a Skolem sequence is defined in the same way, but instead permutes the sequence 0, 0, 1, 1, ..., ''n'' − 1, ''n'' − 1. Example A Langford pairing for ''n'' = 3 is given by the sequence 2, 3, 1, 2, 1, 3. Properties Langford pairings exist only when ''n'' is congruent to 0 or 3 modulo 4; for instance, there is no Langford pairing when ''n'' = 1, 2, or 5. The numbers of different Langford pairings for ''n'' = 1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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