In mathematics, Stirling numbers arise in a variety of analytic and combinatorics problems. They are named after James Stirling, who introduced them in the 18th century. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind.
Several different notations for the Stirling numbers are in use. Stirling numbers of the first kind are written with a small s, and those of the second kind with a large S. The Stirling numbers of the second kind are never negative, but those of the first kind can be negative; hence, there are notations for the "unsigned Stirling numbers of the first kind", which are the Stirling numbers without their signs. Common notations are:
for the ordinary (signed) Stirling numbers of the first kind,
for the unsigned Stirling numbers of the first kind, and
for the Stirling numbers of the second kind.
Abramowitz and Stegun use an uppercase S and a blackletter S, respectively, for the first and second kinds of Stirling number. The notation of brackets and braces, in analogy to the binomial coefficients, was introduced in 1935 by Jovan Karamata and promoted later by Donald Knuth. (The bracket notation conflicts with a common notation for the 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.
Stirling numbers of the first kind
The Stirling numbers of the first kind are the coefficients in the expansion
(Confusingly, the Pochhammer symbol that many use for falling factorials is used in special functions for rising factorials.)
The unsigned Stirling numbers of the first kind,
A few of the Stirling numbers of the first kind are illustrated by the table below:
Stirling numbers of the second kind
Stirling numbers of the second kind count the number of ways to partition a set of n elements into k nonempty subsets. They are denoted by or . The sum
is the nth Bell number.
Using falling factorials, we can characterize the Stirling numbers of the second kind by the identity
The Lah numbers are sometimes called Stirling numbers of the third kind. For example, see Jozsef Sándor and Borislav Crstici, Handbook of Number Theory II, Volume 2.
The Stirling numbers of the first and second kinds can be considered inverses of one another:
where is the Kronecker delta. These two relationships may be understood to be matrix inverse relationships. That is, let s be the lower triangular matrix of Stirling numbers of the first kind, whose matrix elements The inverse of this matrix is S, the lower triangular matrix of Stirling numbers of the second kind, whose entries are 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.
A generalization of the inversion relationship gives the link with the Lah numbers
with the conventions and if .
Abramowitz and Stegun give the following symmetric formulae that relate the Stirling numbers of the first and second kind.
- Bell polynomials
- Cycles and fixed points
- Lah number
- Pochhammer symbol
- Polynomial sequence
- Stirling transform
- Touchard polynomials
- M. Abramowitz and I. Stegun (Eds.). Stirling Numbers of the First Kind., §24.1.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 824, 1972.
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- Victor Adamchik, "On Stirling Numbers and Euler Sums", Journal of Computational and Applied Mathematics 79 (1997) pp. 119–130.
- Arthur T. Benjamin, Gregory O. Preston, Jennifer J. Quinn, A Stirling Encounter with Harmonic Numbers, (2002) Mathematics Magazine, 75 (2) pp 95–103.
- Khristo N. Boyadzhiev, Close encounters with the Stirling numbers of the second kind (2012) Mathematics Magazine, 85 (4) pp 252–266.
- Louis Comtet, Valeur de s(n, k), Analyse combinatoire, Tome second (page 51), Presses universitaires de France, 1970.
- Louis Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, Dordrecht-Holland/Boston-U.S.A., 1974.
- Hsien-Kuei Hwang (1995). "Asymptotic Expansions for the Stirling Numbers of the First Kind". Journal of Combinatorial Theory, Series A 71 (2): 343–351. doi:10.1016/0097-3165(95)90010-1.
- D.E. Knuth, Two notes on notation (TeX source).
- Francis L. Miksa (1901–1975), Stirling numbers of the first kind, "27 leaves reproduced from typewritten manuscript on deposit in the UMT File", Mathematical Tables and Other Aids to Computation, vol. 10, no. 53, January 1956, pp. 37–38 (Reviews and Descriptions of Tables and Books, 7[I]).
- Dragoslav S. Mitrinović, Sur les nombres de Stirling de première espèce et les polynômes de Stirling, AMS 11B73_05A19, Publications de la Faculté d'Electrotechnique de l'Université de Belgrade, Série Mathématiques et Physique (ISSN 0522-8441), no. 23, 1959 (5.V.1959), pp. 1–20.
- John J. O'Connor and Edmund F. Robertson, James Stirling (1692–1770), (September 1998).
- Sixdeniers, J. M.; Penson, K. A.; Solomon, A. I. (2001). "Extended Bell and Stirling Numbers From Hypergeometric Exponentiation". Journal of Integer Sequences 4: 01.1.4..
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- "Sloane's A008275 : Stirling numbers of first kind", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- "Sloane's A008277 : Stirling numbers of 2nd kind", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Stirling numbers of the first kind, s(n,k) at PlanetMath.org. .
- Stirling numbers of the second kind, S(n,k) at PlanetMath.org. .