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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 ordered Bell numbers may be expanded out into a double summation.
The Bell number counts the different ... the ordered Bell numbers. ... A different summation formula represents each Bell number as a sum of Stirling numbers of the ...
This number is known as the nth Bell number. Analogously, the ordered Bell numbers can be computed from the ... this formula is a special case of the kth ...
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 ...
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 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
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