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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 ...
To see this, first note that there are 2 n ordered pairs of complementary subsets A and B. In one case, A is empty, and in another B is empty, so 2 n − 2 ordered pairs of subsets remain. Finally, since we want unordered pairs rather than ordered pairs we divide this last number by 2, giving the result above.
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 .
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:
Order (Latin: ordo) is one of the eight major hierarchical taxonomic ranks in Linnaean taxonomy. It is classified between family and class . In biological classification , the order is a taxonomic rank used in the classification of organisms and recognized by the nomenclature codes .
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.