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Necklace (combinatorics)
In combinatorics, a ''k''-ary necklace of length ''n'' is an equivalence class of ''n''-character strings over an alphabet of size ''k'', taking all rotations as equivalent. It represents a structure with ''n'' circularly connected beads which have ''k'' available colors. A ''k''-ary bracelet, also referred to as a turnover (or free) necklace, is a necklace such that strings may also be equivalent under reflection. That is, given two strings, if each is the reverse of the other, they belong to the same equivalence class. For this reason, a necklace might also be called a fixed necklace to distinguish it from a turnover necklace. Formally, one may represent a necklace as an orbit of the cyclic group acting on ''n''-character strings over an alphabet of size ''k'', and a bracelet as an orbit of the dihedral group. One can count these orbits, and thus necklaces and bracelets, using Pólya's enumeration theorem. Equivalence classes Number of necklaces There are :N_k(n) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficients
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 gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and especia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proofs Of Fermat's Little Theorem
This article collects together a variety of proofs of Fermat's little theorem, which states that :a^p \equiv a \pmod p for every prime number ''p'' and every integer ''a'' (see modular arithmetic). Simplifications Some of the proofs of Fermat's little theorem given below depend on two simplifications. The first is that we may assume that is in the range . This is a simple consequence of the laws of modular arithmetic; we are simply saying that we may first reduce modulo . This is consistent with reducing a^p modulo , as one can check. Secondly, it suffices to prove that :a^ \equiv 1 \pmod p for in the range . Indeed, if the previous assertion holds for such , multiplying both sides by yields the original form of the theorem, :a^p \equiv a \pmod p On the other hand, if , the theorem holds trivially. Combinatorial proofs Proof by counting necklaces This is perhaps the simplest known proof, requiring the least mathematical background. It is an attractive example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necklace Splitting Problem
Necklace splitting is a picturesque name given to several related problems in combinatorics and measure theory. Its name and solutions are due to mathematicians Noga Alon and Douglas B. West. The basic setting involves a necklace with beads of different colors. The necklace should be divided between several partners (e.g. thieves), such that each partner receives the same amount of every color. Moreover, the number of ''cuts'' should be as small as possible (in order to waste as little as possible of the metal in the links between the beads). Variants The following variants of the problem have been solved in the original paper: #Discrete splitting: The necklace has k\cdot n beads. The beads come in t different colors. There are k\cdot a_i beads of each color i, where a_i is a positive integer. Partition the necklace into k parts (not necessarily contiguous), each of which has exactly a_i beads of color ''i''. Use at most (k-1)t cuts. Note that if the beads of each color are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necklace Problem
The necklace problem is a problem in recreational mathematics concerning the reconstruction of necklaces (cyclic arrangements of binary values) from partial information. Formulation The necklace problem involves the reconstruction of a necklace of n beads, each of which is either black or white, from partial information. The information specifies how many copies the necklace contains of each possible arrangement of k black beads. For instance, for k=2, the specified information gives the number of pairs of black beads that are separated by i positions, for i=0,\dots, \lfloor n/2-1 \rfloor . This can be made formal by defining a k-configuration to be a necklace of k black beads and n-k white beads, and counting the number of ways of rotating a k-configuration so that each of its black beads coincides with one of the black beads of the given necklace. The necklace problem asks: if n is given, and the numbers of copies of each k-configuration are known up to some threshold k\le K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversion (discrete Mathematics)
In computer science and discrete mathematics, an inversion in a sequence is a pair of elements that are out of their natural total order, order. Definitions Inversion Let \pi be a permutation. There is an inversion of \pi between i and j if i \pi(j). The inversion is indicated by an ordered pair containing either the places (i, j) or the elements \bigl(\pi(i), \pi(j)\bigr). The #Example:_All_permutations_of_four_elements, inversion set is the set of all inversions. A permutation's inversion set using place-based notation is the same as the Permutation#Definition, inverse permutation's inversion set using element-based notation with the two components of each ordered pair exchanged. Likewise, a permutation's inversion set using element-based notation is the same as the inverse permutation's inversion set using place-based notation with the two components of each ordered pair exchanged. Inversions are usually defined for permutations, but may also be defined for sequences ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutations
In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is &nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class Representative
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyndon Word
In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its rotations. Lyndon words are named after mathematician Roger Lyndon, who investigated them in 1954, calling them standard lexicographic sequences. Anatoly Shirshov introduced Lyndon words in 1953 calling them regular words. Lyndon words are a special case of Hall words; almost all properties of Lyndon words are shared by Hall words. Definitions Several equivalent definitions exist. A k-ary Lyndon word of length n > 0 is an n-character string over an alphabet of size k, and which is the unique minimum element in the lexicographical ordering in the multiset of all its rotations. Being the singularly smallest rotation implies that a Lyndon word differs from any of its non-trivial rotations, and is therefore aperiodic.; . Alternately, a word w is a Lyndon word if and only if it is nonempty and lexicographically stri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Inversion Formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra. Statement of the formula The classic version states that if and are arithmetic functions satisfying : g(n)=\sum_f(d)\quad\textn\ge 1 then :f(n)=\sum_\mu(d)\,g\!\left(\frac\right)\quad\textn\ge 1 where is the Möbius function and the sums extend over all positive divisors of (indicated by d \mid n in the above formulae). In effect, the original can be determined given by using the inversion formula. The two sequences are said to be Möbius transforms of each other. The formula is also correct if and are f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted \mu(x). Definition The Möbius function is defined by :\mu(n) = \begin 1 & \text n = 1 \\ (-1)^k & \text n \text k \text \\ 0 & \text n \text > 1 \end The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where \delta_ is the Kronecker delta, \lambda(n) is the Liouville function, Prime omega function, \omega(n) is the number of distinct prime divisors of n, and Prime omega function, \Omega(n) is the number of prime factors of n, counted with multiplicity. Another characterization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
