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The number on the left hand side of a given row is the Bell number for that row. ( B i ← x i , 1 {\displaystyle B_{i}\leftarrow x_{i,1}} ) Here are the first five rows of the triangle constructed by these rules:
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 ...
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:
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 sum of the subscripts in a monomial is equal to the total number of elements. Thus, the number of monomials that appear in the partial Bell polynomial is equal to the number of ways the integer n can be expressed as a summation of k positive integers. This is the same as the integer partition of n into k parts. For instance, in the above ...
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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.