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| In [[mathematics]], '''Stirling numbers''' arise in a variety of [[Analysis (mathematics)|analytic]] and [[combinatorics]] problems. They are named after [[James Stirling (mathematician)|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]].
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| ==Notation==
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| 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:
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| : <math> s(n,k)\,</math>
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| for the ordinary (signed) Stirling numbers of the first kind,
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| : <math> c(n,k)=\left[{n \atop k}\right]=|s(n,k)|\,</math> | |
| for the unsigned Stirling numbers of the first kind, and
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| : <math> S(n,k)=\left\{\begin{matrix} n \\ k \end{matrix}\right\}= S_n^{(k)} \,</math> | |
| for the Stirling numbers of the second kind.
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| [[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 coefficient]]s.) 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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| ==Stirling numbers of the first kind==
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| {{main|Stirling numbers of the first kind}}
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| The '''Stirling numbers of the first kind''' are the coefficients in the expansion
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| :<math>(x)_{n} = \sum_{k=0}^n s(n,k) x^k.</math> | |
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| where <math>(x)_{n}</math> (a [[Pochhammer symbol]]) denotes the [[falling factorial]],
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| :<math>(x)_{n}=x(x-1)(x-2)\cdots(x-n+1).</math> | |
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| Note that (''x'')<sub>0</sub> = 1 because it is an [[empty product]]. [[Combinatorics|Combinatorialists]] also sometimes use the notation <math style="vertical-align:baseline;">x^{\underline{n\!}}</math> for the falling factorial, and <math style="vertical-align:baseline;">x^{\overline{n\!}}</math> for the rising factorial.<ref>{{cite book|last=Aigner|first=Martin|title=A Course In Enumeration|publisher=Springer|year=2007|pages=561|chapter=Section 1.2 - Subsets and Binomial Coefficients|isbn=3-540-39032-4}}</ref>
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| (Confusingly, the Pochhammer symbol that many use for ''falling'' factorials is used in [[special function]]s for ''rising'' factorials.)
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| The unsigned Stirling numbers of the first kind,
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| :<math>c(n,k)=\left[{n \atop k}\right]=|s(n,k)|=(-1)^{n-k} s(n,k)</math>
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| (with a lower-case "''s''"), count the number of [[permutation]]s of ''n'' elements with ''k'' disjoint [[cyclic permutation|cycle]]s.
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| ==Stirling numbers of the second kind==
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| {{main|Stirling numbers of the second kind}}
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| '''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 <math>S(n,k)</math> or <math>\textstyle \lbrace{n\atop k}\rbrace</math>.<ref>Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) ''[[Concrete Mathematics]]'', Addison-Wesley, Reading MA. ISBN 0-201-14236-8, p. 244.</ref> The sum
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| :<math>\sum_{k=0}^n S(n,k) = B_n</math> | |
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| is the ''n''th [[Bell numbers|Bell number]].
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| Using falling factorials, we can characterize the Stirling numbers of the second kind by the identity
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| :<math>\sum_{k=0}^n S(n,k)(x)_k=x^n.</math> | |
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| ==Lah numbers==
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| {{main|Lah numbers}}
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| The Lah numbers are sometimes called Stirling numbers of the third kind. For example [http://books.google.com/books?id=B2WZkvmFKk8C&pg=PA464&lpg=PA464&dq=%22Stirling+numbers+of+the+third+kind%22&source=bl&ots=JhIJKIhaFH&sig=_0-CWfixhUoAuhh7DAo4fJco6y4&hl=en&ei=BKh2TfnBJ_KH0QGn17XZBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CCAQ6AEwAQ#v=onepage&q=%22Stirling%20numbers%20of%20the%20third%20kind%22&f=false see].
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| ==Inversion relationships==
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| The Stirling numbers of the first and second kinds can be considered to be inverses of one another:
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| :<math>\sum_{n=0}^{\max\{j,k\}} s(n,j) S(k,n) = \delta_{jk}</math>
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| and
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| :<math>\sum_{n=0}^{\max\{j,k\}} S(n,j) s(k,n) = \delta_{jk}</math>
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| where <math>\delta_{jk}</math> 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 first kind, so that it has matrix elements
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| :<math>s_{nk}=s(n,k).\,</math>
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| Then, the [[matrix inverse|inverse]] of this matrix is ''S'', the [[lower triangular matrix]] of Stirling numbers of second kind. Symbolically, one writes
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| :<math>s^{-1} = S\,</math>
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| where the matrix elements of ''S'' are
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| :<math>S_{nk}=S(n,k).</math>
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| Note that 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.
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| A generalization of the inversion relationship gives the link with the Lah numbers <math> L(n,k):</math>
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| :<math> (-1)^n L(n,k) = \sum_{z}(-1)^{z} s(n,z)S(z,k),</math>
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| with the conventions <math>L(0,0)=1</math> and <math>L(n , k )=0</math> if <math>k>n</math>.
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| ==Symmetric formulae==
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| Abramowitz and Stegun give the following symmetric formulae that relate the Stirling numbers of the first and second kind.
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| :<math>s(n,k) = \sum_{j=0}^{n-k} (-1)^j {n-1+j \choose n-k+j} {2n-k \choose n-k-j} S(n-k+j,j)</math>
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| and
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| :<math>S(n,k) = \sum_{j=0}^{n-k} (-1)^j {n-1+j \choose n-k+j} {2n-k \choose n-k-j} s(n-k+j,j).</math>
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| == See also ==
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| * [[Bell polynomials]]
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| * [[Cycles and fixed points]]
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| * [[Lah number]]
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| * [[Pochhammer symbol]]
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| * [[Polynomial sequence]]
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| * [[Stirling transform]]
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| * [[Touchard polynomials]]
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| ==References==
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| {{Reflist}}
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| * 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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| * Milton Abramowitz and Irene A. Stegun, eds., [http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP ''Handbook of Mathematical Functions (with Formulas, Graphs and Mathematical Tables)''], U.S. Dept. of Commerce, National Bureau of Standards, Applied Math. Series 55, 1964, 1046 pages (9th Printing: November 1970) - Combinatorial Analysis, Table 24.4, Stirling Numbers of the Second Kind (author: Francis L. Miksa), p. 835.
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| * Victor Adamchik, "[http://www-2.cs.cmu.edu/~adamchik/articles/stirling.pdf On Stirling Numbers and Euler Sums]", Journal of Computational and Applied Mathematics '''79''' (1997) pp. 119–130.
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| * Arthur T. Benjamin, Gregory O. Preston, Jennifer J. Quinn, ''[http://www.math.hmc.edu/~benjamin/papers/harmonic.pdf A Stirling Encounter with Harmonic Numbers]'', (2002) Mathematics Magazine, '''75''' (2) pp 95–103.
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| * Khristo N. Boyadzhiev, ''Close encounters with the Stirling numbers of the second kind'' (2012) Mathematics Magazine, '''85''' (4) pp 252–266.
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| * Louis Comtet, [http://www.techniques-ingenieur.fr/page/af202niv10002/permutations.html#2.2 ''Valeur de ''s''(''n'', ''k'')''], Analyse combinatoire, Tome second (page 51), Presses universitaires de France, 1970.
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| * Louis Comtet, ''Advanced Combinatorics: The Art of Finite and Infinite Expansions'', Reidel Publishing Company, Dordrecht-Holland/Boston-U.S.A., 1974.
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| * {{cite journal| author=Hsien-Kuei Hwang |title=Asymptotic Expansions for the Stirling Numbers of the First Kind |journal=Journal of Combinatorial Theory, Series A |volume=71 |issue=2 |pages=343–351 |year=1995 |url=http://citeseer.ist.psu.edu/577040.html |doi=10.1016/0097-3165(95)90010-1}}
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| * [[Donald Knuth|D.E. Knuth]], [http://www-cs-faculty.stanford.edu/~knuth/papers/tnn.tex.gz ''Two notes on notation''] (TeX source).
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| * Francis L. Miksa (1901–1975), [http://links.jstor.org/sici?sici=0891-6837%28195601%2910%3A53%3C35%3ARADOTA%3E2.0.CO%3B2-X ''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]).
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| * Dragoslav S. Mitrinović, [http://pefmath2.etf.bg.ac.rs/files/23/23.pdf ''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.
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| * John J. O'Connor and Edmund F. Robertson, [http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Stirling.html ''James Stirling (1692–1770)''], (September 1998).
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| * {{cite journal| first1=J. M. |last1=Sixdeniers |first2= K. A. |last2=Penson
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| |first3=A. I. |last3= Solomon | url = http://www.cs.uwaterloo.ca/journals/JIS/VOL4/SIXDENIERS/bell.pdf
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| |title= Extended Bell and Stirling Numbers From Hypergeometric Exponentiation
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| |year=2001
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| |journal = Journal of Integer Sequences | volume= 4 | pages=01.1.4}}.
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| * {{cite news| first1=Michael Z. | last1=Spivey | title=Combinatorial sums and finite differences
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| |doi=10.1016/j.disc.2007.03.052 | journal=Discr. Math.
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| |year=2007 | volume=307 | number=24 | pages=3130–3146}}
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| * {{SloanesRef |sequencenumber=A008275|name=Stirling numbers of first kind}}
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| * {{SloanesRef |sequencenumber=A008277|name=Stirling numbers of 2nd kind}}
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| * {{planetmath reference |id=2809|title=Stirling numbers of the first kind, s(n,k)}}.
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| * {{planetmath reference |id=2805|title=Stirling numbers of the second kind, S(n,k)}}.
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| [[Category:Permutations]]
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| [[Category:Q-analogs]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Integer sequences]]
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