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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
The ordered Bell numbers were studied in the 19th century by Arthur Cayleyand 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.
Bell polynomials. In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in Faà di Bruno's formula.
Dobiński's formula. In combinatorial mathematics, Dobiński's formula[1] states that the n -th Bell number Bn (i.e., the number of partitions of a set of size n) equals. where denotes Euler's number. The formula is named after G. Dobiński, who published it in 1877.
Bell state measurement. The Bell measurement is an important concept in quantum information science: It is a joint quantum-mechanical measurement of two qubits that determines which of the four Bell states the two qubits are in. Quantum circuit that performs Bell decoding. Bell states are sometimes called EPR pairs.
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In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by or . [1] Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions.
For numberings of the set of computable functions, see Numbering (computability theory). In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his ...