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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 ...
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.
Each coefficient is a polynomial in the cumulants; these are the Bell polynomials, named after Eric Temple Bell. [citation needed] This sequence of polynomials is of binomial type. In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants.
The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that that is, the characteristic polynomial is an annihilating polynomial for. One use for the Cayley–Hamilton theorem is that it allows An to be expressed as a linear combination of the lower matrix powers of A: When the ring is ...
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 ...
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1 1 , 3 2 , 7 5 , 17 12 , and 41 29 , so the sequence of Pell numbers begins ...