Stirling numbers of the first kind

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one).

(The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers in general.)

The original definition of Stirling numbers of the first kind was algebraic. These numbers, usually written s(n, k), are signed integers whose sign, positive or negative, depends on the parity of n − k. Afterwards, the absolute values of these numbers, |s(n, k)|, which are known as unsigned Stirling numbers of the first kind, were found to count permutations, so in combinatorics the (signed) Stirling numbers of the first kind, s(n, k), are often defined as counting numbers multiplied by a sign factor. That is the approach taken on this page.

Most identities on this page are stated for unsigned Stirling numbers. Note that the notations on this page are not universal.

The unsigned Stirling numbers of the first kind are denoted in various ways by different authors. Common notations are ${\displaystyle c(n,k),}$ ${\displaystyle |s(n,k)|}$ and ${\displaystyle \left[{n \atop k}\right]}$. (The last is also common notation for the Gaussian coefficients.) They count the number of permutations of n elements with k disjoint cycles. For example, of the ${\displaystyle 3!=6}$ permutations of three elements, there is one permutation with three cycles (the identity permutation, given in one-line notation by ${\displaystyle 123}$ or in cycle notation by ${\displaystyle (1)(2)(3)}$), three permutations with two cycles (${\displaystyle 132=(1)(23)}$, ${\displaystyle 213=(3)(12)}$, and ${\displaystyle 321=(2)(13)}$) and two permutations with one cycle (${\displaystyle 312=(132)}$ and ${\displaystyle 231=(123)}$). Thus, ${\displaystyle \left[{3 \atop 3}\right]=1}$, ${\displaystyle \left[{3 \atop 2}\right]=3}$ and ${\displaystyle \left[{3 \atop 1}\right]=2}$.

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