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: they are the coefficients s(n, k) in the expansion of the falling factorial into powers of the variable x: For example, ( x ) 3 = x ( x − 1 ) ( x − 2 ) = 1 ⋅ x 3 − 3 ⋅ x 2 + 2 ⋅ x {displaystyle (x)_{3}=x(x-1)(x-2)=1cdot x^{3}-3cdot x^{2}+2cdot x} , 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} . Subsequently, it was discovered that the absolute values |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} . They may be defined directly to be the number of permutations of n elements with k disjoint cycles. For example, of the 3 ! = 6 {displaystyle 3!=6} permutations of three elements, there is one permutation with three cycles (the identity permutation, given in one-line notation by 123 {displaystyle 123} or in cycle notation by ( 1 ) ( 2 ) ( 3 ) {displaystyle (1)(2)(3)} ), three permutations with two cycles ( 132 = ( 1 ) ( 23 ) {displaystyle 132=(1)(23)} , 213 = ( 12 ) ( 3 ) {displaystyle 213=(12)(3)} , and 321 = ( 13 ) ( 2 ) {displaystyle 321=(13)(2)} ) and two permutations with one cycle ( 312 = ( 132 ) {displaystyle 312=(132)} and 231 = ( 123 ) {displaystyle 231=(123)} ). Thus, [ 3 3 ] = 1 {displaystyle left=1} , [ 3 2 ] = 3 {displaystyle left=3} and [ 3 1 ] = 2 {displaystyle left=2} . These can be seen to agree with the previous calculation of s ( n , k ) {displaystyle s(n,k)} for n = 3 {displaystyle n=3} . The unsigned Stirling numbers may also be defined algebraically, as the coefficients of the rising factorial: 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} is also common notation for the Gaussian coefficients.)