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Bell cites several earlier publications on these numbers, beginning with Dobiński 1877 which gives Dobiński's formula for the Bell numbers. Bell called these numbers "exponential numbers"; the name "Bell numbers" and the notation B n for these numbers was given to them by Becker & Riordan 1948. [29]
In mathematics, the Bell triangle is a triangle of numbers analogous to Pascal's triangle, whose values count partitions of a set in which a given element is the largest singleton. It is named for its close connection to the Bell numbers , [ 1 ] which may be found on both sides of the triangle, and which are in turn named after Eric Temple Bell .
In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the weak orderings on a set of elements. Weak orderings arrange their elements into a sequence allowing ties , such as might arise as the outcome of a horse race .
This number is known as the nth Bell number. ... Calculator for Stirling Numbers of the Second Kind; Set Partitions: Stirling Numbers; Jack van der Elsen (2005).
The Bell numbers are repeated along both sides of this triangle. The numbers within the triangle count partitions in which a given element is the largest singleton. The number of partitions of an n-element set into exactly k (non-empty) parts is the Stirling number of the second kind S(n, k).
The total number of monomials appearing in a complete Bell polynomial B n is thus equal to the total number of integer partitions of n. Also the degree of each monomial, which is the sum of the exponents of each variable in the monomial, is equal to the number of blocks the set is divided into.
In combinatorial mathematics, Dobiński's formula [1] states that the th Bell number, the number of partitions of a set of size , equals = =!, where denotes Euler's number. The formula is named after G. Dobiński, who published it in 1877.
The value at 1 of the nth Touchard polynomial is the nth Bell number, i.e., the number of partitions of a set of size n: =.If X is a random variable with a Poisson distribution with expected value λ, then its nth moment is E(X n) = T n (λ), leading to the definition: