In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the unsigned 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.
Definitions
Definition by algebra The Stirling numbers of the first kind are the coefficients s ( n , k ) {\displaystyle s(n,k)} in the expansion of the falling factorial
( x ) n = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) {\displaystyle (x)_{n}=x(x-1)(x-2)\cdots (x-n+1)}
into powers of the variable x {\displaystyle x} :
( x ) n = ∑ k = 0 n s ( n , k ) x k , {\displaystyle (x)_{n}=\sum _{k=0}^{n}s(n,k)x^{k},}
For example, ( x ) 3 = x ( x − 1 ) ( x − 2 ) = x 3 − 3 x 2 + 2 x {\displaystyle (x)_{3}=x(x-1)(x-2)=x^{3}-3x^{2}+2x} , leading to the values s ( 3 , 3 ) = 1 {\displaystyle s(3,3)=1} , s ( 3 , 2 ) = − 3 {\displaystyle s(3,2)=-3} , and s ( 3 , 1 ) = 2 {\displaystyle s(3,1)=2} . By the empty product convention, ( x ) 0 = 1 {\displaystyle (x)_{0}=1} and therefore s ( 0 , 0 ) = 1 {\displaystyle s(0,0)=1} . The unsigned Stirling numbers may also be defined algebraically as the coefficients of the rising factorial:
x n ¯ = x ( x + 1 ) ⋯ ( x + n − 1 ) = ∑ k = 0 n [ n k ] x k {\displaystyle x^{\overline {n}}=x(x+1)\cdots (x+n-1)=\sum _{k=0}^{n}\left[{n \atop k}\right]x^{k}} . The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources; the square bracket notation [ n k ] {\displaystyle \left[{n \atop k}\right]} is also common notation for the Gaussian coefficients.
Definition by permutation Subsequently, it was discovered that the absolute values | s ( n , k ) | {\displaystyle |s(n,k)|} of these numbers are equal to the number of permutations of certain kinds. These absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted c ( n , k ) {\displaystyle c(n,k)} or [ n k ] {\displaystyle \left[{n \atop k}\right]} . They may be defined directly to be the number of permutations of n {\displaystyle n} elements with k {\displaystyle k} disjoint cycles.
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