Concept
Stirling number
Related concepts (12)
Generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) 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.
Stirling numbers of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). 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 first kind.
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 or . 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.
Binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type).
Bell number
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s. The Bell numbers are denoted , where is an integer greater than or equal to zero. Starting with , the first few Bell numbers are 1, 1, 2, 5, 15, 52, 203, 877, 4140, ... .
Bell polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's formula. The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by where the sum is taken over all sequences j1, j2, j3, ..., jn−k+1 of non-negative integers such that these two conditions are satisfied: The sum is called the nth complete exponential Bell polynomial.
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 The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)_n , where n is a non-negative integer.
Twelvefold way
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. The idea of the classification is credited to Gian-Carlo Rota, and the name was suggested by Joel Spencer. Let N and X be finite sets. Let and be the cardinality of the sets. Thus N is an n-set, and X is an x-set.
Stirling polynomials
In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence considered below most notably including the Sheffer sequence form of the sequence, , defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, , which also satisfy a characteristic ordinary generating function and that are of use in generalizing the Stirling numbers (of both kinds) to arbitrary complex-valued inputs.
List of factorial and binomial topics
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). 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 numbe