Nº de Bell B(n) (n≤10)

The Bell Numbers B(n) calculator for n up to 10 provides the number of ways to partition a set with n elements. Bell numbers are a fundamental sequence in combinatorics, representing the total number of partitions of a set, where a partition is a division of the set into non-empty, disjoint subsets. For example, B(3) = 5, since a set of 3 elements can be partitioned in 5 distinct ways.

It works based on a precomputed table of Bell numbers for n from 0 to 10. The user inputs n (between 0 and 10) and the calculator returns the corresponding Bell number. The values come from the known sequence: B(0)=1, B(1)=1, B(2)=2, B(3)=5, B(4)=15, B(5)=52, B(6)=203, B(7)=877, B(8)=4140, B(9)=21147, B(10)=115975.

When to use: this calculator is useful for counting problems in combinatorics, such as determining the number of possible equivalence relations on a finite set, or the number of ways to group elements without internal order. It is also applied in set theory, probability and statistics, and combinatorial optimization problems. For example, analyzing the number of possible customer groupings into market segments.

Cautions: the calculation is straightforward for n up to 10, but for larger values the numbers grow rapidly and may require recursive implementations or more complex formulas. Ensure n is within the allowed range. The calculator does not provide the partitions themselves, only the total count. For n=0, the empty set has exactly one partition (the empty set).