Stirling number
Numbers describing coefficients in polynomial expansions and partitions.
Stirling numbers are mathematical constructs that arise in analytic and combinatorial problems. They are named after James Stirling, who introduced them in an algebraic context in his 1730 work *Methodus differentialis*. They were later rediscovered and given a combinatorial interpretation by Masanobu Saka in his 1782 text *Sanpō-Gakkai*. Two distinct sets of numbers share this name: Stirling numbers of the first kind and Stirling numbers of the second kind. A third set, Lah numbers, are sometimes called Stirling numbers of the third kind. A common property across all three kinds is that they serve as coefficients relating three different sequences of polynomials important in combinatorics. Each can also be defined as counting the number of partitions of n elements into k non-empty subsets, where each subset has a specific type of order—either no order, cyclical order, or linear order. Several notations exist for these numbers. The signed Stirling numbers of the first kind are often denoted with lowercase s, while the unsigned version, counting permutations of n elements with k disjoint cycles, has its own notation. The Stirling numbers of the second kind, counting partitions of a set into k nonempty subsets, are also commonly notated. The bracket and brace notation, analogous to binomial coefficients, was introduced by Jovan Karamata in 1935 and later promoted by Donald Knuth. The numbers express coefficients in expansions of falling and rising factorials as polynomials. For example, the falling factorial expands with signed Stirling numbers of the first kind as coefficients, while the rising factorial expands with unsigned Stirling numbers of the first kind. Stirling numbers of the second kind provide the reverse relations. These numbers also act as change-of-basis coefficients between polynomial bases, such as the standard basis, falling factorials, and rising factorials. The Stirling numbers of the first and second kinds are matrix inverses of one another. Lah numbers, as Stirling numbers of the third kind, similarly express changes between falling and rising factorials.
- field
- Mathematics
- known_for
- Stirling numbers of the first and second kind
Lore & Background
Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind. Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind. A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of n elements into k non-empty subsets, where each subset is endowed with a certain kind of order (no order, cyclical, or linear).
Introduced by James Stirling in his 1730 work *Methodus differentialis*, these numbers were originally presented in an algebraic context. They were later rediscovered and given a combinatorial interpretation by Masanobu Saka in 1782. The two primary kinds serve as coefficients in the expansion of falling and rising factorials as polynomials. Specifically, signed Stirling numbers of the first kind appear when expanding a falling factorial, while unsigned Stirling numbers of the first kind appear in the expansion of a rising factorial. Stirling numbers of the second kind express the reverse relations, connecting these factorial bases back to standard polynomial powers. These relationships allow the numbers to act as change-of-basis coefficients between three polynomial sequences: standard powers, falling factorials, and rising factorials. The matrices formed by the Stirling numbers of the first and second kinds are inverses of one another. Lah numbers, sometimes called Stirling numbers of the third kind, similarly relate falling and rising factorials. Various notations exist for these numbers, including bracket and brace notations introduced by Jovan Karamata in 1935 and later promoted by Donald Knuth, as well as notations used by Abramowitz and Stegun. The numbers are useful for computing sums of polynomials evaluated at consecutive integers, where expressing a polynomial in the basis of falling factorials simplifies the calculation.
Reader's Guide
Stirling numbers are significant in mathematics because they provide coefficients for expansions of falling and rising factorials as polynomials, and they express relations between powers and factorial polynomials. The signed Stirling numbers of the first kind, denoted s(n,k), appear in the expansion of the falling factorial (x)_n = x(x-1)...(x-n+1) as a sum of powers of x. The unsigned Stirling numbers of the first kind, denoted by bracket notation [n k] or c(n,k), count the number of permutations of n elements with k disjoint cycles and appear in the expansion of the rising factorial x^(n) = x(x+1)...(x+n-1). Stirling numbers of the second kind, denoted S(n,k) or {n k}, count the number of ways to partition a set of n elements into k nonempty subsets and express powers x^n as sums of falling factorials.
Did You Know?
- Lah numbers are sometimes referred to as Stirling numbers of the third kind.
Frequently Asked Questions
What are Stirling number's powers/role?
These numbers encode the coefficients that appear in polynomial expansions and count the ways a set can be broken into non-empty groups. They sit at the intersection of analytic and combinatorial mathematics, serving as a bridge between the two.
Why is Stirling number important?
They are foundational counting tools in combinatorics, capturing how many ways a collection can be partitioned or how polynomial coefficients transform under change of basis. Their utility spans both discrete and analytic branches of mathematics.
What are the different kinds of Stirling numbers?
There are Stirling numbers of the first kind and of the second kind, each counting a different combinatorial structure. Lah numbers are occasionally referred to as Stirling numbers of the third kind, though that label is less standard.
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