Stirlingszámokkal

Stirlingszámokkal, often translated as Stirling numbers, are a family of combinatorial numbers that appear in various areas of mathematics, particularly in number theory and combinatorics. There are two main types of Stirling numbers: the first kind and the second kind.

Stirling numbers of the first kind, denoted by s(n, k) or [n k], count the number of permutations of n elements that have exactly k cycles. These numbers can be defined by a recurrence relation: s(n, k) = s(n-1, k-1) - (n-1)s(n-1, k), with base cases s(n, 0) = 0 for n > 0, s(0, 0) = 1, and s(n, k) = 0 if k > n. The unsigned Stirling numbers of the first kind, |s(n, k)| or (n k), count the permutations with k cycles, while the signed numbers have alternating signs based on parity.

Stirling numbers of the second kind, denoted by S(n, k) or {n k}, count the number of ways to partition a set of n distinct objects into k non-empty, unordered subsets. Their recurrence relation is S(n, k) = S(n-1, k-1) + kS(n-1, k), with base cases S(n, 0) = 0 for n > 0, S(0, 0) = 1, and S(n, k) = 0 if k > n. These numbers are always non-negative.

Both types of Stirling numbers are closely related to the powers of x. Specifically, x^n can be expressed as a sum involving Stirling numbers of the second kind, and the falling factorial x^(n) can be expressed as a sum involving Stirling numbers of the first kind. They have applications in calculus, finite differences, and the enumeration of various combinatorial objects.