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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] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Quarterly
The ''Fibonacci Quarterly'' is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year. It is the primary publication of The Fibonacci Association, which has published it since 1963. Its founding editors were Verner Emil Hoggatt Jr. and Alfred Brousseau; by Clark Kimberling the present editor is Professor Curtis Cooper of the Mathematics Department of the . The ''Fibonacci Quarterly'' has an editorial board of nineteen members and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hockey Stick
A hockey stick is a piece of sports equipment used by the players in all the forms of hockey to move the ball or puck (as appropriate to the type of hockey) either to push, pull, hit, strike, flick, steer, launch or stop the ball/ puck during play with the objective being to move the ball/puck around the playing area using the stick, and then trying to score. The word "stick" is a very generic term for the equipment since the different disciplines of hockey require significant differences in both the form and the size of the stick used for it to be effective in the different sports. Field/ice/roller hockey all have a visually similar form of stick with a long shaft or handle which can be held with two hands, and a curved and flattened end; the end and curvature of these sticks are generally the most visible differences between the sticks for these sports. A modern underwater hockey stick bears little resemblance to any field/ice/roller hockey stick, since it is much smaller to ena ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christmas Stocking
A Christmas stocking is an empty sock or sock-shaped bag that is hung on Saint Nicholas Day or Christmas Eve so that Saint Nicholas (or the related figures of Santa Claus and Father Christmas) can fill it with small toys, candy, fruit, coins or other small gifts when he arrives. These small items are often referred to as stocking stuffers or stocking fillers. The tradition of the Christmas stocking is thought to originate from the life of Saint Nicholas. In some Christmas stories, the contents of the Christmas stocking are the only toys the child receives at Christmas from Santa Claus; in other stories (and in tradition), some presents are also wrapped up in wrapping paper and placed under the Christmas tree. Tradition in Western culture threatens that a child who behaves badly during the year will receive only a piece or pile of coal. Some people even put their Christmas stocking by their bedposts so Santa Claus can fill it by the bed while they sleep. History The origin of the C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Identity
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers ''n'' and ''k'', + = , where \tbinom is a binomial coefficient; one interpretation of the coefficient of the term in the expansion of . There is no restriction on the relative sizes of and , since, if the value of the binomial coefficient is zero and the identity remains valid. Pascal's rule can also be viewed as a statement that the formula \frac = = solves the linear two-dimensional difference equation N_ = N_ + N_, \quad N_ = N_ = 1 over the natural numbers. Thus, Pascal's rule is also a statement about a formula for the numbers appearing in Pascal's triangle. Pascal's rule can also be generalized to apply to multinomial coefficients. Combinatorial proof Pascal's rule has an intuitive combinatorial meaning, that is clearly expressed in this counting proof. ''Proof''. Recall that \tbinom equals the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telescoping Series
In mathematics, a telescoping series is a series whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums only consists of two terms of (a_n) after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences. For example, the series :\sum_^\infty\frac (the series of reciprocals of pronic numbers) simplifies as :\begin \sum_^\infty \frac & = \sum_^\infty \left( \frac - \frac \right) \\ & = \lim_ \sum_^N \left( \frac - \frac \right) \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack = 1. \end An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, ''De dimensione parabolae''. In general Telescoping sums are finite sums in which pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stars And Bars (combinatorics)
In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", "balls and bars", and "dots and dividers") is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting problems, such as how many ways there are to put indistinguishable balls into distinguishable bins. Statements of theorems The stars and bars method is often introduced specifically to prove the following two theorems of elementary combinatorics concerning the number of solutions to an equation. Theorem one For any pair of positive integers and , the number of -tuples of positive integers whose sum is is equal to the number of -element subsets of a set with elements. For example, if and , the theorem gives the number of solutions to (with ) as the binomial coefficient :\binom = \binom = \binom = 84. Theorem two For any pair of positive integers and , t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Counting (proof Technique)
In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which call "one of the most important tools in combinatorics", one describes a finite set from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other. Examples Multiplication (of natural numbers) commutes This is a simple example of double counting, often used when teaching multiplication to young children. In this context, multiplication of natural numbers is introduced as repeated addition, and is then shown to be commutative by counting, in two different ways, a number of items arranged in a rectangular grid. Suppose the grid has n rows and m columns. We first count the items by summing n rows of m items each, then a second time by summing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vandermonde's Identity
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: :=\sum_^r for any nonnegative integers ''r'', ''m'', ''n''. The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie.See for the history. There is a ''q''-analog to this theorem called the ''q''-Vandermonde identity. Vandermonde's identity can be generalized in numerous ways, including to the identity : = \sum_ \cdots . Proofs Algebraic proof In general, the product of two polynomials with degrees ''m'' and ''n'', respectively, is given by :\biggl(\sum_^m a_ix^i\biggr) \biggl(\sum_^n b_jx^j\biggr) = \sum_^\biggl(\sum_^r a_k b_\biggr) x^r, where we use the convention that ''ai'' = 0 for all integers ''i'' > ''m'' and ''bj'' = 0 for all integers ''j'' > ''n''. By the binomial theorem, :(1+x)^ = \sum_^ x^r. U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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