Bell number - Another view of Bell numbers

In combinatorial mathematics, the nth Bell number, named in honor of Eric Temple Bell, is the number of partitions of a set with n members, or equivalently, the number of equivalence relations on it. Starting with B0 = B1 = 1, the first few Bell numbers are (sequence A000110 in OEIS): 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975 (See also breakdown by number of subsets/equivalence classes.) Bell number - Partitions of a set. I ...

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• Bell number - Partitions of a set
• Bell number - Another view of Bell numbers
• Bell number - Properties of Bell numbers
• Bell number - Triangle scheme for calculating Bell numbers

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The Bell numbers can easily be calculated by creating the so-called Bell triangle, also called Aitken's array or the Peirce triangle: Start with the number one. Put this on a row by itself. Start a new row with the rightmost element from the previous row as the leftmost number Determine the numbers not on the left column by taking the sum of the number to the left and the number above the number to the left (the number diagonally up and left of the number we are calculating) Repeat ...

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Bell number, Bell number - Partitions of a set, Bell number - Another view of Bell numbers, Bell number - Properties of Bell numbers, Bell number - Triangle scheme for calculating Bell numbers

Read more here: » Bell number: Encyclopedia II - Bell number - Triangle scheme for calculating Bell numbers

The Bell numbers satisfy this recursion formula: They also satisfy "Dobinski's formula": the n-th moment of a Poisson distribution with expected value 1. And they satisfy "Touchard's congruence": If p is any prime number then Each Bell number is a sum of "Stirling numbers of the second kind" The Stirling number S(n, k) is the number of ways to partition a set of ca ...

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Bell number, Bell number - Partitions of a set, Bell number - Another view of Bell numbers, Bell number - Properties of Bell numbers, Bell number - Triangle scheme for calculating Bell numbers

Read more here: » Bell number: Encyclopedia II - Bell number - Properties of Bell numbers

In general, Bn is the number of partitions of a set of size n. A partition of a set S is defined as a set of nonempty, pairwise disjoint subsets of S whose union is S. For example, B3 = 5 because the 3-element set {a, b, c} can be partitioned in 5 distinct ways: {{a}, {b}, {c}} {{a}, {b, c}} {{b}, {a, c}} {{c}, {a, b ...

See also:

Bell number, Bell number - Partitions of a set, Bell number - Another view of Bell numbers, Bell number - Properties of Bell numbers, Bell number - Triangle scheme for calculating Bell numbers

Read more here: » Bell number: Encyclopedia II - Bell number - Partitions of a set