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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]
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: T n ( 1 ) = B n . {\displaystyle T_{n}(1)=B_{n}.} If X is a random variable with a Poisson distribution with expected value λ, then its n th moment is E( X n ) = T n (λ), leading to the definition:
The total number of these faces is 1 + 14 + 36 + 24 = 75, an ordered Bell number, corresponding to the summation formula above for =. [17] By expanding each Stirling number in this formula into a sum of binomial coefficients, the formula for the
The total number of partitions of an n-element set is the Bell number B n. The first several Bell numbers are B 0 = 1, B 1 = 1, B 2 = 2, B 3 = 5, B 4 = 15, B 5 = 52, and B 6 = 203 (sequence A000110 in the OEIS). Bell numbers satisfy the recursion + = = and have the exponential generating function
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 relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 but not 5 ≥ 7. The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1 ...
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.