Catalan numbers

Fig 5 Various ways of generating third Catalan number ... 

The Catalan numbers C-N (rarely called Segner numbers after Johann Andreas von Segner (1704—1777))144 can also be found in the On-Line Encyclopedia of Integer Sequences-see sequence A000108 1, 1, 2, 5, 14, 42, 132, 429, 1430,. .. or they can be generated using the following formula ... 

The Catalan numbers could have also been called the Euler numbers because they were discovered by Leonhard Euler (1707-1783) for counting... 

Tal Kubo from the Mathematics Department at Harvard University is one of the world s leading experts on this batrachion. He notes that the sequence is subtly connected with a range of seemingly unrelated topics in mathematics variants of Pascal s triangle, the Gaussian distribution, combinatorial operations on finite sets, and Catalan numbers. Tal Kubo and Ravi Vakil have developed algorithms to compute the behavior of the batrachion as it nears infinity. Indeed, they have found that the... 

The Catalan numbers are counting the valid numbers of right and left brackets. For example, there are exactly five legal strings of three pairs of brackets ... 

The pattern of numbers in the left part of Table 3.3 has a close relationship with the Pascal triangle and can be viewed as a vertically truncated Pascal-like triangle. The numbers appearing in the last two columns (which are identical) are known as the Catalan numbers ... 

On p 882 are listed the first few Catalan numbers. The Cs was misprinted as 1030 instead of the correct value, 1430. 

Fig. 1 Results from a redundancy analysis of stream nutrient concentrations vs. catchment characteristics in 31 headwater catchments in Catalonia (Spain). Note how phosphorus (SRP) and ammonium (NH4) appear associated with urban point and diffuse sources (Urban percent urban land use, Inhabs number of inhabitants, eq. inhab inhabitant equivalents), whereas nitrate (NO3) concentration is positively associated with irrigated agriculture (IrrAg) and bovine cattle units (ECU), and negatively with mean catchment slope (slope) and percent forest land (Forest). The other variables are Area log area, P precipitation, NonIrrAg non irrigated agriculture. Data are from the Catalan water agency (http //ww.gencat.cat/ac)...

Some Contributions in English The Arbor Scientiae A New Encyclopedia in the Thirteenth Century Occitan-Catalan Cultural Context / Lola Badia — The Structure of the Arbor Scientiae / Anthony Bonner — The Role of Numbers in the Structure of the Arbor... 

FIG. 3.3. Polyhedra (a) the regular solids, (b) some Archimedean semi-regular solids, (c) some Catalan semi-regular solids. (The numbers are the numbers of vertices.)... 

The number of edges and symmetry of each Catalan polyhedron can be found in the entry of the corresponding Archimedean dual in Table 3. 

Catalonia is undoubtedly the leading situation, more recently followed by a growing number of schemes in the Basque Country, based on the same operational model (originally borrowed from North Italy). The Catalan development took its implementation from a regional law (Law 6/93) that stipulated compulsory programmes for the source segregation of organic waste in all municipalities with a population over 5,000 inhabitants, which was then extended to all municipalities. 

They occur in several problems of interest in chem-istry, 2 ° particularly in enumeration of so-called excited valence structures. They are named after the Belgian mathematician Eugene Charles Catalan (1814-1894), but the 18th century Mongolian scientist Ming Antu had already known these numbers. ... 

Yet another inorganic network includesthe Catalan fluorite network [11], the structure of a number of mineral fluorides including CaF, shown in Fig. 11. 

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