in mathematics, stirling numbers arise in a variety of analytic and combinatorics problems. they are named after james stirling, who introduced them in the 18th century. two different sets of numbers bear this name: the stirling numbers of the first kind and the stirling numbers of the second kind. ==notation== several different notations for the stirling numbers are in use. stirling numbers of the first kind are written with a small s, and those of the second kind with a large s. the stirling numbers of the second kind are never negative, but those of the first kind can be negative; hence, there are notations for the "unsigned stirling numbers of the first kind", which are the stirling numbers without their signs. common notations are: : s(n,k)\, for the ordinary (signed) stirling numbers of the first kind, : c(n,k)=\left[{n \atop k}\right]=|s(n,k)|\, for the unsigned stirling numbers of the first kind, and : s(n,k)=\left\{\begin{matrix} n \\ k \end{matrix}\right\}= s_n^{(k)} \, for the stirling numbers of the second kind. abramowitz and stegun use an uppercase s and a blackletter s, respectively, for the first and second kinds of stirling number. the notation of brackets and braces, in analogy to the binomial coefficients, was introduced in 1935 by jovan karamata and promoted later by donald knuth. (the bracket notation conflicts with a common notation for the gaussian coefficients.) the mathematical motivation for this type of notation, as well as additional stirling number formulae, may be found on the page for stirling numbers and exponential generating functions. ==stirling numbers of the first kind==  the stirling numbers of the first kind are the coefficients in the expansion :(x)_{n} = \sum_{k=0}^n s(n,k) x^k. where (x)_{n} (a pochhammer symbol) denotes the falling factorial, :(x)_{n}=x(x-1)(x-2)\cdots(x-n+1). note that (x)0 = 1 because it is an empty product. combinatorialists also sometimes use the notation x^{\underline{n\!}} for the falling factorial, and x^{\overline{n\!}} for the rising factorial. (confusingly, the pochhammer symbol that many use for falling factorials is used in special functions for rising factorials.) the unsigned stirling numbers of the first kind, :c(n,k)=\left[{n \atop k}\right]=|s(n,k)|=(-1)^{n-k} s(n,k) (with a lower-case "s"), count the number of permutations of n elements with k disjoint cycles. a few of the stirling numbers of the first kind are illustrated by the table below: :\begin{array}{ccccccccccc} &&&
Factorial_and_binomial_topics Integer_sequences Permutations Q-analogs