In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus differentialis (1730). They were rediscovered and given a combinatorial meaning by Masanobu Saka in his 1782 Sanpō-Gakkai (The Sea of Learning on Mathematics).

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. Each kind is detailed in its respective article, this one serving as a description of relations between them.

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).

Notation

Several different notations for Stirling numbers are in use. Ordinary (signed) Stirling numbers of the first kind are commonly denoted by

Unsigned Stirling numbers of the first kind, which count the number of permutations of n elements with k disjoint cycless, are denoted by

Stirling numbers of the second kind, which count the number of ways to partition a set of n elements into k nonempty subsets:

Abramowitz and Stegun and a blackletter, respectively, for the first and second kinds of Stirling number. The notation of brackets and braces, in analogy to binomial coefficients, was introduced in 1935 by Jovan Karamata and promoted later by Donald Knuth, though the bracket notation conflicts with a common notation for Gaussian coefficients. The mathematical motivation for this type of notation, as well as additional Stirling number formulae, may be found on the page for Stirling numbers and exponential generating functions.

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Expansions of falling and rising factorials

Stirling numbers express coefficients in expansions of falling and rising factorials as polynomials.

That is, the falling factorial is a polynomial in of degree whose expansion is

with (signed) Stirling numbers of the first kind as coefficients.

by convention, because it is an empty product for the rising factorial are also often used. (Confusingly, the Pochhammer symbol that many use for falling factorials is used in special functions for rising factorials.)

Similarly, the rising factorial is a polynomial in of degree whose expansion is

Stirling numbers of the second kind express the reverse relations:

and

As change of basis coefficients

Considering the set of polynomials in the (indeterminate) variable x as a vector space, each of the three sequences

is a basis. That is, every polynomial in x (similarly for the other two bases). The above relations then express the change of basis between them, as summarized in the following commutative diagram:

The coefficients for the two bottom changes are described by the Lah numbers with falling and rising factorials as above.

Falling factorials define, up to scaling, the same polynomials as binomial coefficients are thus described by similar formulas:

1. .

Example

Expressing a polynomial in the basis of falling factorials is useful for calculating sums of the polynomial evaluated at consecutive integers. Indeed, the sum of falling factorials with fixed k)

This can be proved by induction.

For example, the sum of fourth powers of integers up to n (this time with n included), is:

Here the Stirling numbers can be computed from their definition as the number of partitions of 4 elements into k non-empty unlabeled subsets.

in the standard basis is given by Faulhaber's formula, which in general is more complicated.

As inverse matrices

The Stirling numbers of the first and second kinds can be considered inverses of one another:

and

is the Kronecker delta. These two relationships may be understood to be matrix inverse relationships. That is, let s be the lower triangular matrix The inverse of this matrix is S, the lower triangular matrix Symbolically, this is written

Although s and S are infinite, so calculating a product entry involves an infinite sum, the matrix multiplications work because these matrices are lower triangular, so only a finite number of terms in the sum are nonzero.

Lah numbers

are sometimes called Stirling numbers of the third kind..

These numbers are coefficients expressing falling factorials in terms of rising factorials and vice versa:

, completing the diagram. In particular, one formula is the inverse of the other, thus:

:

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Enumeratively can be defined as the number of partitions of n elements into k non-empty unlabeled subsets, where each subset is endowed with no order, a cyclic order, or a linear order, respectively. In particular, this implies the inequalities:

Inversion relations and the Stirling transform

, related by a finite sum Stirling number formula given by

, we have a corresponding inversion formula given by

.

These inversion relations between the two sequences translate into functional equations between the sequence exponential generating functions given by the Stirling (generating function) transform as

and

, the differential operators:

Another pair of "inversion" relations involving the Stirling numbers relate the forward differences derivatives by the formulas

Similar properties

Table of similarities

See the specific articles for details.

Symmetric formulae

Abramowitz and Stegun give the following symmetric formulae that relate the Stirling numbers of the first and second kind.

and

Stirling numbers with negative integral values

The Stirling numbers can be extended to negative integral values, but not all authors do so in the same way. Regardless of the approach taken, it is worth noting that Stirling numbers of first and second kind are connected by the relations:

when n and k:

Donald Knuth defined the more general Stirling numbers by extending a recurrence relation are zero if n is negative and k is nonnegative, or if n is nonnegative and k is negative, and so we have, for any integers n and k,

On the other hand, for positive integers n and k, David Branson). In this approach, one has the following extension of the recurrence relation of the Stirling numbers of the first kind:

1. ,

for negative integral n.

is a Bell number.

.

See also

• Bell polynomials

• Catalan number

• Cycles and fixed points

• Pochhammer symbol

• Polynomial sequence

• Touchard polynomials

• Stirling permutation

Citations

References

Further reading