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Specifiable Combinatorial Class
In combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is mostly associated with Philippe Flajolet and is detailed in Part A of his book with Robert Sedgewick, ''Analytic Combinatorics'', while the rest of the book explains how to use complex analysis in order to get asymptotic and probabilistic results on the corresponding generating functions. During two centuries, generating functions were popping up via the corresponding recurrences on their coefficients (as can be seen in the seminal works of Bernoulli, Euler, Arthur Cayley, Schröder, Ramanujan, Riordan, Knuth, , etc.). It was then slowly realized that the generating functions were capturing many other facets of the initial discrete combinatorial objects, and that this could be done in a more direct formal way: The recursive nature of some combinatorial structures translates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Species
In combinatorics, combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for deriving the generating functions of discrete structures, which allows one to not merely count these structures but give bijective proof, bijective proofs involving them. Examples of combinatorial species are (wiktionary:finite, finite) Graph (discrete mathematics), graphs, permutations, Tree (graph theory), trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size. One goal of species theory is to be able to analyse complicated structures by describing them in terms of transformations and combinations of simpler structures. These operations correspond to equivalent manipulations of generating functions, so producing such functions for complicated structures is much easier than with other methods. The theory was introduced, carefully elaborated and applied by Canadian researchers around André ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Permutation Statistics
The statistics of random permutations, such as the cycle structure of a random permutation are of fundamental importance in the analysis of algorithms, especially of sorting algorithms, which operate on random permutations. Suppose, for example, that we are using quickselect (a cousin of quicksort) to select a random element of a random permutation. Quickselect will perform a partial sort on the array, as it partitions the array according to the pivot. Hence a permutation will be less disordered after quickselect has been performed. The amount of disorder that remains may be analysed with generating functions. These generating functions depend in a fundamental way on the generating functions of random permutation statistics. Hence it is of vital importance to compute these generating functions. The article on random permutations contains an introduction to random permutations. The fundamental relation Permutations are sets of labelled cycles. Using the labelled case of the Fla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Labelled Enumeration Theorem
In combinatorial mathematics, the labelled enumeration theorem is the counterpart of the Pólya enumeration theorem for the labelled case, where we have a set of labelled objects given by an exponential generating function (EGF) ''g''(''z'') which are being distributed into ''n'' slots and a permutation group ''G'' which permutes the slots, thus creating equivalence classes of configurations. There is a special re-labelling operation that re-labels the objects in the slots, assigning labels from 1 to ''k'', where ''k'' is the total number of nodes, i.e. the sum of the number of nodes of the individual objects. The EGF f_n(z) of the number of different configurations under this re-labelling process is given by :f_n(z) = \frac. In particular, if ''G'' is the symmetric group of order ''n'' (hence, , ''G'', = ''n''!), the functions f_n(z) can be further combined into a single generating function In mathematics, a generating function is a representation of an infinite seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Generating Function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Index
In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used in combinatorial enumeration. Each permutation π of a finite set of objects partitions that set into cycles; the cycle index monomial of π is a monomial in variables ''a''1, ''a''2, … that describes the cycle type of this partition: the exponent of ''a''''i'' is the number of cycles of π of size ''i''. The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of ''cycle index''. Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group's action. This is the main ingredient in the Pólya ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Generating Function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...s of a formal power series. Generating functions are often expressed in Closed-form expression, closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pólya Enumeration Theorem
The Pólya enumeration theorem, also known as the Redfield–Pólya theorem and Pólya counting, is a theorem in combinatorics that both follows from and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. The theorem was first published by J. Howard Redfield in 1927. In 1937 it was independently rediscovered by George Pólya, who then greatly popularized the result by applying it to many counting problems, in particular to the enumeration of chemical compounds. The Pólya enumeration theorem has been incorporated into symbolic combinatorics and the theory of combinatorial species. Simplified, unweighted version Let ''X'' be a finite set and let ''G'' be a group of permutations of ''X'' (or a finite symmetry group that acts on ''X''). The set ''X'' may represent a finite set of beads, and ''G'' may be a chosen group of permutations of the beads. For example, if ''X'' is a necklace of ''n'' beads in a circle, then rotational symmetry is r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Class
In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size. Counting sequences and isomorphism The ''counting sequence'' of a combinatorial class is the sequence of the numbers of elements of size ''i'' for ''i'' = 0, 1, 2, ...; it may also be described as a generating function that has these numbers as its coefficients. The counting sequences of combinatorial classes are the main subject of study of enumerative combinatorics. Two combinatorial classes are said to be isomorphic if they have the same numbers of objects of each size, or equivalently, if their counting sequences are the same.. Frequently, once two combinatorial classes are known to be isomorphic, a bijective proof of this equivalence is sought; such a proof may be interpreted as showing that the objects in the two isomorphic classes are cryptomorphic to eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating expression (mathematics), mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes ''exact'' computation with expressions containing variable (mathematics), variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called ''computer algebra systems'', with the term ''system'' alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Generation
In computing, procedural generation is a method of creating data algorithmically as opposed to manually, typically through a combination of human-generated content and algorithms coupled with computer-generated randomness and processing power. In computer graphics, it is commonly used to create textures and 3D models. In video games, it is used to automatically create large amounts of content in a game. Depending on the implementation, advantages of procedural generation can include smaller file sizes, larger amounts of content, and randomness for less predictable gameplay. Overview The term ''procedural'' refers to the process that computes a particular function. Fractals are geometric patterns which can often be generated procedurally. Commonplace procedural content includes textures and meshes. Sound is often also procedurally generated, and has applications in both speech synthesis as well as music. It has been used to create compositions in various genres of electronic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
