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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 Faà di Bruno's formula.
A 1951 paper by H. D. Block and H. P. Thielman sparked interest in the subject of fixed points of commuting functions. [1] Building on earlier work by J. F. Ritt and A. G. Walker, Block and Thielman identified sets of pairwise commuting polynomials and studied their properties, including that all of the polynomials in each set would share a common fixed point.
The Bell numbers are named after Eric Temple Bell, who wrote about them in 1938, following up a 1934 paper in which he studied the Bell polynomials. [27] [28] Bell did not claim to have discovered these numbers; in his 1938 paper, he wrote that the Bell numbers "have been frequently investigated" and "have been rediscovered many times". Bell ...
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 . [1] Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions.
The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by. where is a Stirling number of the second kind, i.e., the number of partitions of a set of size n into k disjoint non-empty subsets. [1][2][3][4]
The number of alternative assignments for a given number of workers, taking into account the choices of how many stages to use and how to assign workers to each stage, is an ordered Bell number. [29] As another example, in the computer simulation of origami, the ordered Bell numbers give the number of orderings in which the creases of a crease ...
Stirling numbers of the first kind are the coefficients in the expansion of the falling factorial. into powers of the variable : For example, , leading to the values , , and . Subsequently, it was discovered that the absolute values of these numbers are equal to the number of permutations of certain kinds. These absolute values, which are known ...
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ...
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