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Stirling Transform
In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by :b_n=\sum_^n \left\ a_k, where \left\ is the Stirling number of the second kind, also denoted ''S''(''n'',''k'') (with a capital ''S''), which is the number of partitions of a set of size ''n'' into ''k'' parts. The inverse transform is :a_n=\sum_^n s(n,k) b_k, where ''s''(''n'',''k'') (with a lower-case ''s'') is a Stirling number of the first kind. Berstein and Sloane (cited below) state "If ''a''''n'' is the number of objects in some class with points labeled 1, 2, ..., ''n'' (with all labels distinct, i.e. ordinary labeled structures), then ''b''''n'' is the number of objects with points labeled 1, 2, ..., ''n'' (with repetitions allowed)." If :f(x) = \sum_^\infty x^n is a formal power series, and :g(x) = \sum_^\infty x^n with ''a''''n'' and ''b''''n'' as above, then :g(x) = f(e^x-1).\, Likewise, the inverse transform leads to the generating function identity : ... [...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] |
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] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...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] |
Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and Notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., ''X'' is a disjoint union of the subsets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said to exhaust or cover ''X''. See also collectively exhaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form a x^n where x^n is the nth power of a variable x (n is a non-negative integer), and a is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the x^n are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Transform
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. Definition The binomial transform, ''T'', of a sequence, , is the sequence defined by :s_n = \sum_^n (-1)^k a_k. Formally, one may write :s_n = (Ta)_n = \sum_^n T_ a_k for the transformation, where ''T'' is an infinite-dimensional operator with matrix elements ''T''''nk''. The transform is an involution, that is, :TT = 1 or, using index notation, :\sum_^\infty T_T_ = \delta_ where \delta_ is the Kronecker delta. The original series can be regained by :a_n=\sum_^n (-1)^k s_k. The binomial transform of a sequence is just the ''n''th forward differences of the sequence, with odd differences carrying a negative sign, namely: :\begin s_0 &= a_0 \\ s_1 &= - (\Delta a) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function Transformation
In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas applied to a sequence generating function (see integral transformations) or weighted sums over the higher-order derivatives of these functions (see derivative transformations). Given a sequence, \_^, the ordinary generating function (OGF) of the sequence, denoted F(z), and the exponential generating function (EGF) of the sequence, denoted \widehat(z), are defined by the formal power series :F(z) = \sum_^\infty f_n z^n = f_0 + f_1 z + f_2 z^2 + \cdots :\widehat(z) = \sum_^\infty \frac z^n = \frac + \frac z + \frac z^2 + \cdots. In this article, we use the convention that the ordinary (exponential) generating function for a sequence \ is denoted by the uppercase function F(z) / \widehat(z) for some fixed or formal z when the context o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Factorial And Binomial Topics
{{Short description, none This is a list of factorial and binomial topics in mathematics. See also binomial (other). * Abel's binomial theorem * Alternating factorial *Antichain *Beta function *Bhargava factorial *Binomial coefficient **Pascal's triangle *Binomial distribution *Binomial proportion confidence interval *Binomial-QMF (Daubechies wavelet filters) *Binomial series *Binomial theorem *Binomial transform *Binomial type *Carlson's theorem *Catalan number **Fuss–Catalan number * Central binomial coefficient *Combination *Combinatorial number system *De Polignac's formula *Difference operator *Difference polynomials *Digamma function *Egorychev method * Erdős–Ko–Rado theorem *Euler–Mascheroni constant *Faà di Bruno's formula *Factorial *Factorial moment *Factorial number system *Factorial prime *Gamma distribution *Gamma function *Gaussian binomial coefficient *Gould's sequence *Hyperfactorial *Hypergeometric distribution * Hypergeometric function identities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial And Binomial Topics
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
