Enharmonic

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.

Notation

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:

$s(n,k)$

for the ordinary (signed) Stirling numbers of the first kind,

$c(n,k)=\left[\genfrac{}{}{0ex}{}{n}{k}\right]=|s(n,k)|$

for the unsigned Stirling numbers of the first kind, and

$S(n,k)=\left\{\begin{array}{c}n\\ k\end{array}\right\}=S_n^{(k)}$

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

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The Stirling numbers of the first kind are the coefficients in the expansion

$(x{)}_n=\sum _{k=0}^{n}s(n,k)x^k.$

where $(x{)}_n$ (a Pochhammer symbol) denotes the falling factorial,

$(x{)}_n=x(x-1)(x-2)\cdots (x-n+1).$

Note that (x)₀ = 1 because it is an empty product. Combinatorialists also sometimes use the notation $x^{\underset{\_}{n}}$ for the falling factorial, and $x^{\bar{n}}$ for the rising factorial.

(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,

$c(n,k)=\left[\genfrac{}{}{0ex}{}{n}{k}\right]=|s(n,k)|=(-1{)}^{n-k}s(n,k)$

(with a lower-case "s"), count the number of permutations of n elements with k disjoint cycles.

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 $S(n,k)$ or $\left\{\genfrac{}{}{0ex}{}{n}{k}\right\}$. The sum

$\sum _{k=0}^{n}S(n,k)=B_n$

is the nth Bell number.

Using falling factorials, we can characterize the Stirling numbers of the second kind by the identity

$\sum _{k=0}^{n}S(n,k)(x{)}_k=x^n.$

Lah numbers

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The Lah numbers are sometimes called Stirling numbers of the third kind. For example see.

Inversion relationships

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

$\sum _{n=0}^{\max \{j,k\}}s(n,j)S(k,n)={\delta }_{jk}$

and

$\sum _{n=0}^{\max \{j,k\}}S(n,j)s(k,n)={\delta }_{jk}$

where ${\delta }_{jk}$ 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

$s_{nk}=s(n,k).$

Then, the inverse of this matrix is S, the lower triangular matrix of Stirling numbers of second kind. Symbolically, one writes

$s^{-1}=S$

where the matrix elements of S are

$S_{nk}=S(n,k).$

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.

A generalization of the inversion relationship gives the link with the Lah numbers $L(n,k):$

$(-1{)}^{n}L(n,k)=\sum _z(-1{)}^{z}s(n,z)S(z,k),$

with the conventions $L(0,0)=1$ and $L(n,k)=0$ if $k>n$.

Symmetric formulae

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

$s(n,k)=\sum _{j=0}^{n-k}(-1{)}^j(\genfrac{}{}{0ex}{}{n-1+j}{n-k+j})(\genfrac{}{}{0ex}{}{2n-k}{n-k-j})S(n-k+j,j)$

and

$S(n,k)=\sum _{j=0}^{n-k}(-1{)}^j(\genfrac{}{}{0ex}{}{n-1+j}{n-k+j})(\genfrac{}{}{0ex}{}{2n-k}{n-k-j})s(n-k+j,j).$

See also

• Bell polynomials
• Cycles and fixed points
• Lah number
• Pochhammer symbol
• Polynomial sequence
• Stirling transform
• Touchard polynomials

References

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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.
• Milton Abramowitz and Irene A. Stegun, eds., 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.
• 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.
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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 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).
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang.
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