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The ordered Bell numbers were studied in the 19th century by Arthur Cayley and William Allen Whitworth. They are named after Eric Temple Bell, who wrote about the Bell numbers, which count the partitions of a set; the ordered Bell numbers count partitions that have been equipped with a total order.
As suggested by the set notation above, the ordering of subsets within the family is not considered; ordered partitions are counted by a different sequence of numbers, the ordered Bell numbers. is 1 because there is exactly one partition of the empty set. This partition is itself the empty set; it can be interpreted as a family of subsets of ...
Definition. A strict weak ordering on a set is a strict ... These numbers are also called the Fubini numbers or ordered Bell numbers. For example, for a set of three ...
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 .
An example of second-order conditioning. In classical conditioning, second-order conditioning or higher-order conditioning is a form of learning in which a stimulus is first made meaningful or consequential for an organism through an initial step of learning, and then that stimulus is used as a basis for learning about some new stimulus.
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:
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The model of hierarchical complexity (MHC) is a formal theory and a mathematical psychology framework for scoring how complex a behavior is. [4] Developed by Michael Lamport Commons and colleagues, [3] it quantifies the order of hierarchical complexity of a task based on mathematical principles of how the information is organized, [5] in terms of information science.