StirlingZahlen

StirlingZahlen, in German often called Stirlingzahlen, are two families of combinatorial numbers named after James Stirling. They arise in counting problems and in the relations between powers, factorials and partitions. The two main families are the Stirling numbers of the first kind and the Stirling numbers of the second kind. They are usually denoted respectively by s(n,k) for the signed first kind, c(n,k) or |s(n,k)| for the unsigned first kind, and S(n,k) for the second kind.

Stirling numbers of the first kind measure the cycle structure of permutations. The unsigned version c(n,k) counts the number of permutations of n elements with exactly k cycles. The signed version satisfies s(0,0)=1, s(n,0)=0 for n>0, and s(n,k) = s(n-1,k-1) - (n-1) s(n-1,k). The relationship c(n,k) = |s(n,k)| holds with s(n,k) = (-1)^{n-k} c(n,k).

Stirling numbers of the second kind count partitions of an n-element set into k nonempty blocks. They satisfy S(0,0)=1, S(n,0)=0 for n>0, and S(n,k) = k S(n-1,k) + S(n-1,k-1). The total number of partitions of an n-element set is the Bell number B_n = sum_{k=0}^n S(n,k).

Connections and generating functions: x^n = sum_{k=0}^n S(n,k) (x)_k, where (x)_k is the falling factorial, and (x)_n = sum_{k=0}^n s(n,k) x^k. Stirling Zahlen thus link ordinary powers, factorials and partitions and have numerous identities and applications in combinatorics and number theory.