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In [[combinatorics|combinatorial mathematics]], the '''Lobb number''' ''L''<sub>''m'',''n''</sub> counts the number of ways that ''n'' + ''m'' open parentheses and ''n'' − ''m'' close parentheses can be arranged to form the start of a valid sequence of [[Dyck language|balanced parentheses]].<ref>{{Cite journal | |||
| last1 = Koshy | |||
| first1 = Thomas | |||
| title = Lobb's generalization of Catalan's parenthesization problem | |||
| journal = [[College Mathematics Journal|The College Mathematics Journal]] | |||
| volume = 40 | |||
| issue = 2 | |||
| pages = 99–107 | |||
|date=March 2009 | |||
| doi = 10.4169/193113409X469532 | |||
}}</ref> | |||
Lobb numbers form a natural generalization of the [[Catalan number]]s, which count the number of complete strings of balanced parentheses of a given length. Thus, the ''n''th Catalan number equals the Lobb number ''L''<sub>0,''n''</sub>.<ref>{{cite book | |||
| last1 = Koshy | |||
| first1 = Thomas | |||
| title= Catalan Numbers with Applications | |||
| year = 2008 | |||
| publisher = Oxford University Press | |||
| isbn = 978-0-19-533454-8 | |||
}}</ref> They are named after Andrew Lobb, who used them to give a simple [[mathematical induction|inductive proof]] of the formula for the ''n''<sup>th</sup> Catalan number.<ref>{{Cite journal | |||
| last1 = Lobb | |||
| first1 = Andrew | |||
| title = Deriving the ''n''th Catalan number | |||
| journal = [[Mathematical Gazette]] | |||
| volume = 83 | |||
| issue = 8 | |||
| pages = 109–110 | |||
|date=March 1999 | |||
}}</ref> | |||
The Lobb numbers are parameterized by two non-negative [[integers]] ''m'' and ''n'' with ''n'' ≥ ''m'' ≥ 0. The (''m'', ''n'')<sup>th</sup> Lobb number ''L''<sub>''m'',''n''</sub> is given in terms of [[binomial coefficient]]s by the formula | |||
:<math>L_{m,n} = \frac{2m+1}{m+n+1}\binom{2n}{m+n} \qquad\text{ for }n \ge m \ge 0.</math> | |||
As well as counting sequences of parentheses, the Lobb numbers also count the number of ways in which ''n'' + ''m'' copies of the value +1 and ''n'' − ''m'' copies of the value −1 may be arranged into a sequence such that all of the [[partial sum]]s of the sequence are non-negative. | |||
==References== | |||
{{reflist}} | |||
{{Classes of natural numbers}} | |||
{{DEFAULTSORT:Lobb Numbers}} | |||
[[Category:Integer sequences]] | |||
[[Category:Factorial and binomial topics]] | |||
[[Category:Enumerative combinatorics]] | |||
Revision as of 22:34, 2 June 2013
In combinatorial mathematics, the Lobb number Lm,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses.[1]
Lobb numbers form a natural generalization of the Catalan numbers, which count the number of complete strings of balanced parentheses of a given length. Thus, the nth Catalan number equals the Lobb number L0,n.[2] They are named after Andrew Lobb, who used them to give a simple inductive proof of the formula for the nth Catalan number.[3]
The Lobb numbers are parameterized by two non-negative integers m and n with n ≥ m ≥ 0. The (m, n)th Lobb number Lm,n is given in terms of binomial coefficients by the formula
As well as counting sequences of parentheses, the Lobb numbers also count the number of ways in which n + m copies of the value +1 and n − m copies of the value −1 may be arranged into a sequence such that all of the partial sums of the sequence are non-negative.
References
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Template:Classes of natural numbers
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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
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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
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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