Stirling number
Numbers describing coefficients in polynomial expansions and partitions.
Stirling numbers are mathematical constructs that arise in analytic and combinatorial problems. Two sets bear this name: Stirling numbers of the first kind and Stirling numbers of the second kind, with Lah numbers sometimes referred to as Stirling numbers of the third kind.
- 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).
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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