{"history":[],"article":{"id":"07dfdfdb-1207-4162-8d1b-96a446875547","title":"Stirling numbers of the first kind","slug":"stirling-numbers-of-the-first-kind","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), especially in [combinatorics](https://bliptext.com/articles/combinatorics), **Stirling numbers of the first kind** arise in the study of permutations.\nIn particular, the unsigned Stirling numbers of the first kind count [permutations](https://bliptext.com/articles/permutation) according to their number of [cycles](https://bliptext.com/articles/cycles-and-fixed-points) (counting [fixed points](https://bliptext.com/articles/fixed-point-mathematics) as [cycles](https://bliptext.com/articles/cycles-and-fixed-points) of length one).\nThe Stirling numbers of the first and [second kind](https://bliptext.com/articles/stirling-numbers-of-the-second-kind) can be understood as inverses of one another when viewed as [triangular matrices](https://bliptext.com/articles/triangular-matrix).\nThis article is devoted to specifics of Stirling numbers of the first kind.\nIdentities linking the two kinds appear in the article on [Stirling numbers](https://bliptext.com/articles/stirling-number).\n\n# Definitions\n\n## Definition by algebra\nSigned Stirling numbers of the first kind are the coefficients s(n,k) in the expansion of the [falling factorial](https://bliptext.com/articles/falling-factorial)\ninto powers of the variable x:\nFor example,, leading to the values s(3, 3) = 1, , and s(3, 1) = 2.\nThe unsigned Stirling numbers may also be defined algebraically as the coefficients of the [rising factorial](https://bliptext.com/articles/rising-factorial):\nThe notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources; the square bracket notation is also common notation for the [Gaussian coefficients](https://bliptext.com/articles/gaussian-binomial-coefficient).\n\n## Definition by permutation\nSubsequently, it was discovered that the absolute values |s(n,k)| of these numbers are equal to the number of [permutations](https://bliptext.com/articles/permutation) of certain kinds.\nThese absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted c(n,k) or.\nThey may be defined directly to be the number of [permutations](https://bliptext.com/articles/permutation) of n elements with k disjoint [cycles](https://bliptext.com/articles/cyclic-permutation).\nFor example, of the 3!\n= 6 permutations of three elements, there is one permutation with three cycles (the [identity permutation](https://bliptext.com/articles/identity-permutation), given in [one-line notation](https://bliptext.com/articles/permutation) by 123 or in [cycle notation](https://bliptext.com/articles/cycle-notation) by (1)(2)(3)), three permutations with two cycles (,, and ) and two permutations with one cycle (312 = (132) and 231 = (123)).\nThus, and.\nThese can be seen to agree with the previous algebraic calculations of s(n, k) for n = 3.\nFor another example, the image at right shows that : the [symmetric group](https://bliptext.com/articles/symmetric-group) on 4 objects has 3 permutations of the form\nand 8 permutations of the form\nThese numbers can be calculated by considering the orbits as [conjugacy classes](https://bliptext.com/articles/conjugacy-class).\n[Alfréd Rényi](https://bliptext.com/articles/alfr-d-r-nyi) observed that the unsigned Stirling number of the first kind also counts the number of permutations of size n with k left-to-right maxima.\n# [s(4,2)=11 | upload.wikimedia.org/wikipedia/commons/e/ed/Stirling///number///of///the///first///kind///s%284%2C2%29.svg]\n\n## Signs\nThe signs of the signed Stirling numbers of the first kind depend only on the parity of\nn − k\n.\n\n# Recurrence relation\nThe unsigned Stirling numbers of the first kind follow the [recurrence relation](https://bliptext.com/articles/recurrence-relation)\nfor k > 0, with the boundary conditions\nfor n>0.\nIt follows immediately that the signed Stirling numbers of the first kind satisfy the recurrence\n\n# Table of values\nBelow is a [triangular array](https://bliptext.com/articles/triangular-array) of unsigned values for the Stirling numbers of the first kind, similar in form to [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle).\nThese values are easy to generate using the recurrence relation in the previous section.\n\n# Properties\n\n## Simple identities\nUsing the [Kronecker delta](https://bliptext.com/articles/kronecker-delta) one has,\nand\nAlso\nand\nSimilar relationships involving the Stirling numbers hold for the [Bernoulli polynomials](https://bliptext.com/articles/bernoulli-polynomials).\nMany relations for the Stirling numbers shadow similar relations on the [binomial coefficients](https://bliptext.com/articles/binomial-coefficient).\nThe study of these 'shadow relationships' is termed [umbral calculus](https://bliptext.com/articles/umbral-calculus) and culminates in the theory of [Sheffer sequences](https://bliptext.com/articles/sheffer-sequences).\nGeneralizations of the [Stirling numbers](https://bliptext.com/articles/stirling-numbers) of both kinds to arbitrary complex-valued inputs may be extended through the relations of these triangles to the [Stirling convolution polynomials](https://bliptext.com/articles/stirling-polynomial).\nNote that all the combinatorial proofs above use either binomials or multinomials of n.\nTherefore if p is prime, then:\nfor 1 0, with the boundary conditions\nfor n>0.\nIt follows immediately that the signed Stirling numbers of the first kind satisfy the recurrence\n\n# Table of values\nBelow is a [triangular array](https://bliptext.com/articles/triangular-array) of unsigned values for the Stirling numbers of the first kind, similar in form to [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle).\nThese values are easy to generate using the recurrence relation in the previous section.\n\n# Properties\n\n## Simple identities\nUsing the [Kronecker delta](https://bliptext.com/articles/kronecker-delta) one has,\nand\nAlso\nand\nSimilar relationships involving the Stirling numbers hold for the [Bernoulli polynomials](https://bliptext.com/articles/bernoulli-polynomials).\nMany relations for the Stirling numbers shadow similar relations on the [binomial coefficients](https://bliptext.com/articles/binomial-coefficient).\nThe study of these 'shadow relationships' is termed [umbral calculus](https://bliptext.com/articles/umbral-calculus) and culminates in the theory of [Sheffer sequences](https://bliptext.com/articles/sheffer-sequences).\nGeneralizations of the [Stirling numbers](https://bliptext.com/articles/stirling-numbers) of both kinds to arbitrary complex-valued inputs may be extended through the relations of these triangles to the [Stirling convolution polynomials](https://bliptext.com/articles/stirling-polynomial).\nNote that all the combinatorial proofs above use either binomials or multinomials of n.\nTherefore if p is prime, then:\nfor 1

History