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The exponential Bell polynomial encodes the information related to the ways a set can be partitioned. For example, if we consider a set {A, B, C}, it can be partitioned into two non-empty, non-overlapping subsets, which are also referred to as parts or blocks, in 3 different ways:
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 ...
Touchard polynomials. 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]
hide. In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field the ...
Eric Temple Bell (7 February 1883 – 21 December 1960) was a Scottish-born mathematician and science fiction writer who lived in the United States for most of his life. He published non-fiction using his given name and fiction as John Taine .
Applied to the monic polynomial + = with all coefficients a k considered as free parameters, this means that every symmetric polynomial expression S(x 1,...,x n) in its roots can be expressed instead as a polynomial expression P(a 1,...,a n) in terms of its coefficients only, in other words without requiring knowledge of the roots.
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 ...
Calculus. Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno (1855, 1857), although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast had ...
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