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Lattice Path
In combinatorics, a lattice path in the -dimensional integer lattice of length with steps in the set , is a sequence of vectors such that each consecutive difference v_i - v_ lies in . A lattice path may lie in any lattice in , but the integer lattice is most commonly used. An example of a lattice path in of length 5 with steps in S = \lbrace (2,0), (1,1), (0,-1) \rbrace is L = \lbrace (-1,-2), (0,-1), (2,-1), (2,-2), (2,-3), (4,-3) \rbrace . North-East lattice paths A North-East (NE) lattice path is a lattice path in \mathbb^2 with steps in S = \lbrace (0,1), (1,0) \rbrace . The (0,1) steps are called North steps and denoted by N 's; the (1,0) steps are called East steps and denoted by E 's. NE lattice paths most commonly begin at the origin. This convention allows us to encode all the information about a NE lattice path L in a single permutation word. The length of the word gives us the number of steps of the lattice path, k . The order of the N 's an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Path In Z2
Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornamental pattern of crossing strips of pastry Companies * Lattice Engines, a technology company specializing in business applications for marketing and sales * Lattice Group, a former British gas transmission business * Lattice Semiconductor, a US-based integrated circuit manufacturer Science, technology, and mathematics Mathematics * Lattice (group), a repeating arrangement of points ** Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure ** Lattice (module), a module over a ring which is embedded in a vector space over a field ** Lattice graph, a graph that can be drawn within a repeating arrangement of points ** Lattice-based cryptography, encryption systems based ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schröder Number
In mathematics, the Schröder number S_n, also called a ''large Schröder number'' or ''big Schröder number'', describes the number of lattice paths from the southwest corner (0,0) of an n \times n grid to the northeast corner (n,n), using only single steps north, (0,1); northeast, (1,1); or east, (1,0), that do not rise above the SW–NE diagonal. The first few Schröder numbers are :1, 2, 6, 22, 90, 394, 1806, 8558, ... . where S_0=1 and S_1=2. They were named after the German mathematician Ernst Schröder. Examples The following figure shows the 6 such paths through a 2 \times 2 grid: Related constructions A Schröder path of length n is a lattice path from (0,0) to (2n,0) with steps northeast, (1,1); east, (2,0); and southeast, (1,-1), that do not go below the x-axis. The nth Schröder number is the number of Schröder paths of length n. The following figure shows the 6 Schröder paths of length 2. Similarly, the Schröder numbers count the number of ways to divi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squared
A square is a regular quadrilateral with four equal sides and four right angles. Square or Squares may also refer to: Mathematics and science *Square (algebra), multiplying a number or expression by itself *Square (cipher), a cryptographic block cipher *Global square, a principle in infinitary combinatorics *Square number, an integer that is the square of another integer * Square of a graph *Square wave, a non-sinusoidal periodic waveform Construction * Square (area), an Imperial unit of floor area and other construction materials *Square, a public meeting place: **Garden square, an open space with buildings surrounding a garden **Market square, an open area where market stalls are traditionally set out for trading **Town square, an open area commonly found in the heart of a traditional town used for community gatherings *Square (tool), an L- or T-shaped tool: **Combination square, a tool with a ruled blade and one or more interchangeable heads ** Machinist square, a metalworking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NE Nodes SVG
NE, Ne or ne may refer to: Arts and entertainment * Neutral Evil, an alignment in the American role-playing game ''Dungeons & Dragons'' * New Edition, an American vocal group * Nicomachean Ethics, a collection of ten books by Greek philosopher Aristotle Businesses and organizations * National Express, an English public transport operator * Natural England, an English government agency * New England Patriots, a professional American football team in Foxborough, Massachusetts * New Hope (Macau), a Macau political party * SkyEurope Airlines, a Slovakian airline * New Era Cap Company, an American headwear company Language * Ne (cuneiform), a cuneiform sign * Ne (kana), a Japanese written character * Nepali language * Modern English, sometimes abbreviated NE (to avoid confusion with Middle English) Places * NE postcode area, UK, a postcode for the City of Newcastle upon Tyne, Tyne and Wear * Ne, Liguria, Italy, a ''comune'' in the Province of Genoa * Né (river), a river in southwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walking On Pascal's Triangle SVG
Walking (also known as ambulation) is one of the main gaits of terrestrial locomotion among legged animals. Walking is typically slower than running and other gaits. Walking is defined by an 'inverted pendulum' gait in which the body vaults over the stiff limb or limbs with each step. This applies regardless of the usable number of limbs—even arthropods, with six, eight, or more limbs, walk. Difference from running The word ''walk'' is descended from the Old English ''wealcan'' "to roll". In humans and other bipeds, walking is generally distinguished from running in that only one foot at a time leaves contact with the ground and there is a period of double-support. In contrast, running begins when both feet are off the ground with each step. This distinction has the status of a formal requirement in competitive walking events. For quadrupedal species, there are numerous gaits which may be termed walking or running, and distinctions based upon the presence or absence of a ... [...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 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a ''k''-combination of a Set (mathematics), set ''S'' is a subset of ''k'' distinct elements of ''S''. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has ''n'' elements, the number of ''k''-combinations, denoted as C^n_k, is equal to the binomial coefficient \binom nk = \frac, which can be written using factorials as \textstyle\frac whenever k\leq n, and which is zero when k>n. This formula can be derived from the fact that each ''k''-combination of a set ''S'' of ''n'' members has k! ... [...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] |
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] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
