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Stanley Symmetric Function
In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by in his study of the symmetric group of permutations. Formally, the Stanley symmetric function ''F''''w''(''x''1, ''x''2, ...) indexed by a permutation ''w'' is defined as a sum of certain fundamental quasisymmetric functions. Each summand corresponds to a reduced decomposition of ''w'', that is, to a way of writing ''w'' as a product of a minimal possible number of adjacent transpositions. They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation ''w''0 = ''n''(''n'' − 1)...21 (written here in one-line notation) has exactly : \frac reduced decompositions. (Here \binom denotes the binomial coefficient ''n''(''n'' − 1)/2 and ! denotes the factorial.) Properties The Stanley symmetric function ''F''''w'' is homogeneous with degree e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schubert Polynomials
In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by and are named after Hermann Schubert. Background described the history of Schubert polynomials. The Schubert polynomials \mathfrak_w are polynomials in the variables x_1,x_2,\ldots depending on an element w of the infinite symmetric group S_\infty of all permutations of \N fixing all but a finite number of elements. They form a basis for the polynomial ring \Z _1,x_2,\ldots/math> in infinitely many variables. The cohomology of the flag manifold \text(m) is \Z _1, x_2,\ldots, x_mI, where I is the ideal generated by homogeneous symmetric functions of positive degree. The Schubert polynomial \mathfrak_w is the unique homogeneous polynomial of degree \ell(w) representing the Schubert cycle of w in the cohomology of the flag manifold \text(m) for all sufficiently large m. Properties *If w_0 is the permutation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (having both signs). Whenever not specifically mentioned, this article adheres to the first convention. In some contexts, it makes sense to consider a signed zero (such as floating-point representations of real numbers within computers). In mathematics and physics, the phrase "change of sign" is associated with the generation of the additive inverse (negation, or multiplication by −1) of any object that allows for this construction, and is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate other binary aspects of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Polynomials
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials. Definition (Jacobi's bialternant formula) Schur polynomials are indexed by integer partitions. Given a partition , where , and each is a non-negative integer, the functions a_ (x_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Symmetric Functions
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number ''n'' of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group. The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative. Symmetric polynomials The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, a polynomial is called ''symmetric'' if it stays the same whenever the indeterminates are permuted in any way. More formally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (linear Algebra)
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the ''dimension'' of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Definition A basis of a vector space over a field (such as the real numbers or the complex numbers ) is a linearly independent subset of that spans . This me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversion (discrete Mathematics)
In computer science and discrete mathematics, an inversion in a sequence is a pair of elements that are out of their natural order. Definitions Inversion Let \pi be a permutation. There is an inversion of \pi between i and j if i \pi(j). The inversion is indicated by an ordered pair containing either the places (i, j) or the elements \bigl(\pi(i), \pi(j)\bigr). The inversion set is the set of all inversions. A permutation's inversion set using place-based notation is the same as the inverse permutation's inversion set using element-based notation with the two components of each ordered pair exchanged. Likewise, a permutation's inversion set using element-based notation is the same as the inverse permutation's inversion set using place-based notation with the two components of each ordered pair exchanged. Inversions are usually defined for permutations, but may also be defined for sequences:Let S be a sequence (or multiset permutation). If i S(j), either the pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x^3 + 3 x^2 y + z^7 is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. A form of degree 2 is a quadratic fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. History The term "algebraic combinatorics" was introduced in the late 1970s. Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, ''Algebraic combinatorics'', of the AMS Mathematics Subject Classification, introduced in 1991. Scope Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
