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Catalan's Triangle
In combinatorial mathematics, Catalan's triangle is a number triangle whose entries C(n,k) give the number of strings consisting of ''n'' X's and ''k'' Y's such that no initial segment of the string has more Y's than X's. It is a generalization of the Catalan numbers, and is named after Eugène Charles Catalan. Bailey shows that C(n,k) satisfy the following properties: # C(n,0)=1 \text n\geq 0 . # C(n,1)=n \text n\geq 1 . # C(n+1,k)=C(n+1,k-1)+C(n,k) \text 1 n it is impossible to form a path that does not cross the constraint, i.e. C_(n,k)= 0 . (3) when m\leq k\leq n+m-1 , then C_(n,k) is the number of 'red' paths \left(\begin n+k\\ k \end\right) minus the number of 'yellow' paths that cross the constraint, i.e. \left(\begin (n+m)+(k-m)\\ k-m \end\right) = \left(\begin n+k\\ k-m \end\right). Therefore the number of paths from (0,0) to (k, n) that do not cross the constraint n - k + m - 1 = 0 is as indicated in the formula in the previous section "''Generalization''". Proof 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Triangle
In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index. That is, the ''i''th row contains only ''i'' elements. Examples Notable particular examples include these: *The Bell triangle, whose numbers count the partitions of a set in which a given element is the largest singleton * Catalan's triangle, which counts strings of parentheses in which no close parenthesis is unmatched * Euler's triangle, which counts permutations with a given number of ascents * Floyd's triangle, whose entries are all of the integers in order * Hosoya's triangle, based on the Fibonacci numbers * Lozanić's triangle, used in the mathematics of chemical compounds * Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings * Pascal's triangle, whose entries are the binomial coefficients Triangular arrays of integers in which each row is symmetric and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Numbers
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugène Charles Catalan
Eugène Charles Catalan (30 May 1814 – 14 February 1894) was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics. His notable contributions included discovering a periodic minimal surface in the space \mathbb^3; stating the famous Catalan's conjecture, which was eventually proved in 2002; and, introducing the Catalan number to solve a combinatorial problem. Biography Catalan was born in Bruges (now in Belgium, then under Dutch rule even though the Kingdom of the Netherlands had not yet been formally instituted), the only child of a French jeweller by the name of Joseph Catalan, in 1814. In 1825, he traveled to Paris and learned mathematics at École Polytechnique, where he met Joseph Liouville (1833). In December 1834 he was expelled along with most of the students in his year for political reasons; he resumed his studies in January 1835, graduated that summer, and went on to teach at Châlons-sur-Marne. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis François Antoine Arbogast
Louis François Antoine Arbogast (4 October 1759 – 8 April 1803) was a French mathematician. He was born at Mutzig in Alsace and died at Strasbourg, where he was professor. He wrote on series and the derivatives known by his name: he was the first writer to separate the symbols of operation from those of quantity, introducing systematically the operator notation ''DF'' for the derivative of the function ''F''. In 1800, he published a calculus treatise where the first known statement of what is currently known as Faà di Bruno's formula appears, 55 years before the first published paper of Francesco Faà di Bruno on that topic. Biography He was professor of mathematics at the Collège de Colmar and entered a mathematical competition run by the St Petersburg Academy. His entry was to bring him fame and an important place in the history of the development of the calculus. Arbogast submitted an essay to the St Petersburg Academy in which he came down firmly on the side of Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hockey-stick Identity
In combinatorial mathematics, the identity : \sum^n_= \qquad \text n,r\in\mathbb, \quad n\geq r or equivalently, the mirror-image by the substitution j\to i-r: : \sum^_=\sum^_= \qquad \text n,r\in\mathbb, \quad n\geq r is known as the hockey-stick, Christmas stocking identity, boomerang identity, or Chu's Theorem. The name stems from the graphical representation of the identity on Pascal's triangle: when the addends represented in the summation and the sum itself are highlighted, the shape revealed is vaguely reminiscent of those objects (see hockey stick, Christmas stocking). Proofs Generating function proof We have :X^r + X^ + \dots + X^ = \frac Let X=1+x, and compare coefficients of x^r. Inductive and algebraic proofs The inductive and algebraic proofs both make use of Pascal's identity: :=+. Inductive proof This identity can be proven by mathematical induction on n. Base case Let n=r; :\sum^n_ = \sum^r_= = 1 = = . Inductive step Suppose, for some k\in\math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Catalan Number Proof
A general officer is an officer of high rank in the armies, and in some nations' air forces, space forces, and marines or naval infantry. In some usages the term "general officer" refers to a rank above colonel."general, adj. and n.". OED Online. March 2021. Oxford University Press. https://www.oed.com/view/Entry/77489?rskey=dCKrg4&result=1 (accessed May 11, 2021) The term ''general'' is used in two ways: as the generic title for all grades of general officer and as a specific rank. It originates in the 16th century, as a shortening of ''captain general'', which rank was taken from Middle French ''capitaine général''. The adjective ''general'' had been affixed to officer designations since the late medieval period to indicate relative superiority or an extended jurisdiction. Today, the title of ''general'' is known in some countries as a four-star rank. However, different countries use different systems of stars or other insignia for senior ranks. It has a NATO rank scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
