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The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The n-th Catalan number can be expressed directly in terms of the central binomial coefficients by
The cardinal number quīnque ‘five’, with its cognates Old Irish coíc ‘five’, Greek πέντε pénte ‘five’, Sanskrit पञ्च pañca ‘five’, leads back to Proto-Indo-European pénkʷe; the long -ī-, confirmed by preserved -i-in most Romance descendants, must have been transferred from the ordinal quīntus ‘fifth ...
The first row has been interpreted as the prime numbers between 10 and 20 (i.e., 19, 17, 13, and 11), while a second row appears to add and subtract 1 from 10 and 20 (i.e., 9, 19, 21, and 11); the third row contains amounts that might be halves and doubles, though these are inconsistent. [14]
Substituting k = 1 into this formula gives the Catalan numbers and substituting k = 2 into this formula gives the Schröder–Hipparchus numbers. [7] In connection with the property of Schröder–Hipparchus numbers of counting faces of an associahedron, the number of vertices of the associahedron is given by the Catalan numbers.
The Fuss-Catalan represents the number of legal permutations or allowed ways of arranging a number of articles, that is restricted in some way. This means that they are related to the Binomial Coefficient. The key difference between Fuss-Catalan and the Binomial Coefficient is that there are no "illegal" arrangement permutations within Binomial ...
The rule in Catalan is to follow the number with the last letter in the singular and the last two letters in the plural. [24] Most numbers follow the pattern exemplified by vint '20' (20è m sg, 20a f sg, 20ns m pl, 20es f pl), but the first few ordinals are irregular, affecting the abbreviations of the masculine forms. Superscripting is not ...
The amnesty will cover all events related to the Catalan independence drive from 2012 to present day, including a symbolic vote held in 2014 and an independence referendum in 2017, which was ...
Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order m = 1, 2, 3, ... is a number trapezoid whose entries (,) give the number of strings consisting of n X-s and k Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by m or more. [6]