Stirling number

In combinatorics Stirling numbers of the second kind S(n,k) (with a capital "S") count the number of equivalence relations having k equivalence classes defined on a set with n elements. The sum

$B_{n}=\sum _{k=1}^{n}S(n,k)$

is the nth Bell number. If we let

$(x)_{n}=x(x-1)(x-2)\cdots (x-n+1)$

(in particular, (x)₀ = 1 because it is an empty product) be the falling factorial, we can characterize the Stirling numbers of the second kind by

$\sum _{k=1}^{n}S(n,k)(x)_{k}=x^{n}.$

Unsigned Stirling numbers of the first kind s(n,k) (with a lower-case "s") count the number of permutations of n elements with k disjoint cycles.