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André's Problem
In combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set is a permutation (arrangement) of those numbers so that each entry is alternately greater or less than the preceding entry. For example, the five alternating permutations of are: * 1, 3, 2, 4 because 1 2 < 4, * 1, 4, 2, 3 because 1 < 4 > 2 < 3, * 2, 3, 1, 4 because 2 < 3 > 1 < 4, * 2, 4, 1, 3 because 2 < 4 > 1 < 3, and * 3, 4, 1, 2 because 3 < 4 > 1 < 2. This type of permutation was first studied by [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with Index of a subgroup, index 2 and has therefore factorial, ''n''!/2 elements. It is the kernel (algebra), kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian group, abelian if and only if and simple group, simple if and only if or . A5 is the smallest non-abelian simple group, having order of a group, order 60, and thus the smallest non-solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent (trigonometry)
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To ext ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fence (mathematics)
In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations: :acehbdfi \cdots A fence may be finite, or it may be formed by an infinite alternating sequence extending in both directions. The incidence posets of path graphs form examples of fences. A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century. The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are: :1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042. :. The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube. A partially ordered set is series-parallel if and only if it does not have four elements forming ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boustrophedon Transform
In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag- or serpentine-like) manner—as opposed to a "raster scan" sawtooth-like manner. Definition The boustrophedon transform is a numerical, sequence-generating transformation, which is determined by a binary operation such as addition. Generally speaking, given a sequence: (a_0, a_1, a_2, \ldots), the boustrophedon transform yields another sequence: (b_0, b_1, b_2, \ldots), where b_0 is likely defined equivalent to a_0. The entirety of the transformation itself can be visualized (or imagined) as being constructed by filling-out the triangle as shown in Figure 1. Boustrophedon Triangle To fill-out the numerical Isosceles triangle (Figure 1), you start with the input sequence, (a_0, a_1, a_2, \ldots), and place one value (from the input sequence) per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Longest Alternating Subsequence
In combinatorial mathematics, probability, and computer science, in the longest alternating subsequence problem, one wants to find a subsequence of a given sequence in which the elements are in alternating order, and in which the sequence is as long as possible. Formally, if \mathbf = \ is a sequence of distinct real numbers, then the subsequence \ is ''alternating'' (or ''zigzag'' or ''down-up'') if :x_ > x_ \cdots x_\qquad \text \qquad 1\leq i_1 < i_2 < \cdots < i_k \leq n. Similarly, is ''reverse alternating'' (or ''up-down'') if : Note that every sequence of length 1 is both alternating and reverse alternating. Let denote the length (number of terms) of the longest alternating subsequence of . For example, if we consider some of the permutations of the integers 1,2,3,4,5, we have that * |
Stirling Numbers Of The Second Kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \left\. Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. They are named after James Stirling. The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the second kind. Identities linking the two kinds appear in the article on Stirling numbers. Definition The Stirling numbers of the second kind, written S(n,k) or \lbrace\textstyle\rbrace or with other notations, count the number of ways to partition a set of n labelled objects into k nonempty unlabelled subsets. Equivalently, they count the number of different equivalence relations wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Falling And Rising Factorials
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) . \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, — A reprint of the 1950 edition by Chelsea Publishing. rising sequential product, or upper factorial) is defined as \begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) . \end The value of each is taken to be 1 (an empty product) when n=0. These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)_n, where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used (x)_n with yet another meaning, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Numbers
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maclaurin Series
Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin (1878–1915), Australian general * Ian MacLaurin, Baron MacLaurin of Knebworth (b. 1937) * Richard Cockburn Maclaurin (1870–1920), US physicist and educator See also * Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ... in mathematics, a special case of which is the ''Maclaurin series'' * Maclaurin (crater), a crater on the Moon * McLaurin (other) * MacLaren (surname) * McLaren (other) {{surname, Maclaurin Clan MacLaren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Function
In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is even if ''n'' is an even integer, and it is odd if ''n'' is an odd integer. Even functions are those real functions whose graph is self-symmetric with respect to the and odd functions are those whose graph is self-symmetric with respect to the origin. If the domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function. Early history The concept of even and odd functions appears to date back to the early 18th century, with Leonard Euler playing a significant role in their formalization. Euler introduced the concepts of even and odd functions (using Lati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
