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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>
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| 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>
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| 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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Children which have overweight parents have a 75% chance of becoming obese kids. Most parents know what constitutes a healthy diet and they know which regular children are active.Start early and have fun! Many elite athletes began shooting a puck, or swinging a golf club because soon as they could walk. Even when your child isn't destined to be a professional rated athlete, the more active kids are the more likely they might develop into active adults.There are myriad reasons to explain why kids are piling found on the pounds. Earlier this year, Statistics Canada revealed that just eight % of youth aged 6 to 16 meet the suggested amount of at least 50 - 60 minutes of daily moderate to vigorous physical activity.
What weight would I be satisfied with if my ideal is too low? This really is a fat that's not our ideal, however, it would meet us; be inside a healthy weight range plus would be more fair given our present lifestyle. I like to compare my clients' goals to the bmi chart for the healthiest weight ranges for their height. Here's 1 more way to go green, BMI green, that is (The bmi chart is colored coded with green as the healthy weight ranges. Take a guess the color for unhealthiest?) Scroll down for a link to a bmi chart where I give all hyperlinks.
The Body Mass Index formula has been invented a Belgian Polymath, Adolphe Quetelet in the 1830s to the 1850s plus has first been called the Quetelet index. The calculation is the ratio of the weight divided by a height squared. With the innovation of the internet and help from numerous bmi chart men websites and pediatricians, Body Mass Index Calculators can be selected online. You are able to really go online plus look for the calculators plus many websites will be willing to aid you calculate you own Body Mass Index. These services are free. Use services from sites which may give accurate info. Check the ratings found on the websites and take a look for yourself if the info needed in calculating the Body Mass index is comprehensive enough.
Naturally, youngsters should, and must, gain weight from the all-natural procedure of growth, yet several children go beyond which plus put on excess fatty tissue; i.e. they become fat. Obesity is quickly becoming a severe issue with todays kids, partially by the wrong nutrition plus eating too much of the wrong foods, and partially through ignorance on behalf of the parents that have a misconception that puppy fat is a healthy plus general thing.
Yes, smaller people tend to be somewhat wider in proportion to height, than people of average height. (Head sizes of smaller individuals moreover tend to be greater inside proportion to height.) You are able to see that in randomly chosen images of people. What to do about this scaling glitch?
Measure the distance from the floor to the mark you produced found on the paper that bmi chart women is on the wall. Then on a separate sheet of paper write your measurement. Using a fat scale check your weight. Take off a limited pounds off for your dresses plus goods you have inside your pockets.
Plus, theres a superior chance which when her marriage is inside trouble she could have been deprived inside this department for quite some time. When she finds somebody to take all of which frustration out on, there will be no turning back!
All three of these websites offer synonymous yet different diet tools. You will discover which we like one calorie counter plus body mass index tracker more than another. Which site you choose to use is not because significant because being consistent inside a use of it. Tracking the progress, plus holding yourself accountable are the 2 more important keys to weight loss success.