Connections between Bell numbers, Stirling numbers of first and second kind and Partitions. Evaluation of these numbers.

Bell and Stirling numbers of first and second kind tell the number of ways that n objects, or n cycles in the case of Stirling numbers of first kind, can be distributed in k cells. They are usually obtained through recurrence rules. However, recurrence rules only tell how many distributions are possible, not the specific form of each distribution, so they cannot be used to build the distributions themselves.

Here we present a relationship between these distributions and the P(n) and P(n,k) partitions, where P(n,k) represents the partition of n objects in exactly k parts. Such a relationship shows the nature of these distributions and provides a quick and direct way to compute them as well.