Stirlingszámoknak

Stirlingszámoknak, more commonly known as Stirling numbers, are a pair of sequences that count partitions of a set into a specified number of non-empty subsets or cycles. There are two types: Stirling numbers of the first kind and Stirling numbers of the second kind.

Stirling numbers of the first kind, denoted as $s(n, k)$ or $\begin{bmatrix} n \\ k \end{bmatrix}$, count the number of permutations of $n$ elements with exactly $k$ disjoint cycles. The unsigned Stirling numbers of the first kind, $|s(n, k)|$, are often used and represent the same quantity. They satisfy the recurrence relation: $\begin{bmatrix} n \\ k \end{bmatrix} = (n-1) \begin{bmatrix} n-1 \\ k \end{bmatrix} + \begin{bmatrix} n-1 \\ k-1 \end{bmatrix}$, with base cases $\begin{bmatrix} n \\ 0 \end{bmatrix} = 0$ for $n \ge 1$, $\begin{bmatrix} 0 \\ 0 \end{bmatrix} = 1$, and $\begin{bmatrix} n \\ n \end{bmatrix} = 1$.

Stirling numbers of the second kind, denoted as $S(n, k)$ or $\begin{Bmatrix} n \\ k \end{Bmatrix}$, count the number of ways to partition a set of $n$ elements into exactly $k$ non-empty, unordered subsets. They satisfy the recurrence relation: $\begin{Bmatrix} n \\ k \end{Bmatrix} = k \begin{Bmatrix} n-1 \\ k \end{Bmatrix} + \begin{Bmatrix} n-1 \\ k-1 \end{Bmatrix}$, with base cases $\begin{Bmatrix} n \\ 0 \end{Bmatrix} = 0$ for $n \ge 1$, $\begin{Bmatrix} 0 \\ 0 \end{Bmatrix} = 1$, and $\begin{Bmatrix} n \\ n \end{Bmatrix} = 1$.

These numbers appear in various areas of mathematics, including combinatorics, number theory, and the analysis of algorithms. They are closely related to binomial coefficients and factorials, and have connections to other combinatorial objects.