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Steinhaus–Johnson–Trotter Algorithm
The Steinhaus–Johnson–Trotter algorithm or Johnson–Trotter algorithm, also called plain changes, is an algorithm named after Hugo Steinhaus, Selmer M. Johnson and Hale F. Trotter that generates all of the permutations of n elements. Each permutation in the sequence that it generates differs from the previous permutation by swapping two adjacent elements of the sequence. Equivalently, this algorithm finds a Hamiltonian cycle in the permutohedron. This method was known already to 17th-century English change ringers, and calls it "perhaps the most prominent permutation enumeration algorithm". A version of the algorithm can be implemented in such a way that the average time per permutation is constant. As well as being simple and computationally efficient, this algorithm has the advantage that subsequent computations on the permutations that it generates may be sped up because of the similarity between consecutive permutations that it generates.. Algorithm The sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group 4; Cayley Graph 1,2,6 (3D); Steinhaus–Johnson–Trotter
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fabian Stedman
Fabian Stedman (1640–1713) was an English author and a leading figure in the early history of campanology, particularly in the field of method ringing. He had a key role in publishing two books ''Tintinnalogia'' (1668 with Richard Duckworth) and ''Campanalogia'' (1677 – written solely by him) which are the first two publications on the subject. He is also regarded as being a pioneer in the branch of mathematics known as Group theory. Life Fabian Stedman was born in Yarkhill, Herefordshire, the third son to Reverend Francis Stedman. His father Francis Stedman was born in Aston Munslow, Shropshire in 1598, who took Holy Orders at Yarkhill in 1625. Francis had seven children by two wives. The eldest was Francis Junior who followed his father and became Rector of the parish of Stoke Lacy, Herefordshire in 1660. Fabian Stedman was born in 1640 and baptised at Yarkhill Church on 7 December of that year. At the age of 15 he went to London to learn the trade of master printing, appren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Change Ringing
Change ringing is the art of ringing a set of tuned bells in a tightly controlled manner to produce precise variations in their successive striking sequences, known as "changes". This can be by method ringing in which the ringers commit to memory the rules for generating each change, or by call changes, where the ringers are instructed how to generate each change by instructions from a conductor. This creates a form of bell music which cannot be discerned as a conventional melody, but is a series of mathematical sequences. Change ringing originated following the invention of English full-circle tower bell ringing in the early 17th century, when bell ringers found that swinging a bell through a much larger arc than that required for swing-chiming gave control over the time between successive strikes of the clapper. Ordinarily a bell will swing through a small arc only at a set speed governed by its size and shape in the nature of a simple pendulum, but by swinging through a larg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Radix
Mixed radix numeral systems are non-standard positional numeral systems in which the numerical radix, base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the next smaller one, but not by the same factor. Such units are common for instance in measuring time; a time of 32 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, and 500 milliseconds might be expressed as a number of minutes in mixed-radix notation as: ... 32, 5, 7, 45; 15, 500 ... ∞, 7, 24, 60; 60, 1000 or as :32∞577244560.15605001000 In the tabular format, the digits are written above their base, and a semicolon indicates the radix point. In numeral format, each digit has its associated base attached as a subscript, and the radix point is marked by a full stop, full stop or period. The base for each digit is the number of corresponding units that make up the next larger unit. As a consequence there is no b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial Number System
In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, but as place value of digits. By converting a number less than ''n''! to factorial representation, one obtains a sequence of ''n'' digits that can be converted to a permutation of ''n'' elements in a straightforward way, either using them as Lehmer code or as inversion table representation; in the former case the resulting map from integers to permutations of ''n'' elements lists them in lexicographical order. General mixed radix systems were studied by Georg Cantor. The term "factorial number system" is used by Knuth, while the French equivalent "numération factorielle" was first used in 1888. The term "factoradic", which is a portmanteau of factorial and mixed radix, appears to be of more recent date.The term "factoradic" is apparently introduced in . Definitio ... [...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 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 inversion set is the set of all inversions. A permutation's inversion set using place-based notation is the same as the 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:Let S be a sequence (or multiset permutation). If i S(j), either the pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9. In any standard positional numeral system, a number is conventionally written as with ''x'' as the string of digits and ''y'' as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four. Etymology ''Radix'' is a Latin word for "root". ''Root'' can be considered a synonym for ''base,'' in the arithmetical sense. In numeral systems In the system with radix 13, for example, a string of digits such as 398 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gray Code
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representation of the decimal value "1" in binary would normally be "" and "2" would be "". In Gray code, these values are represented as "" and "". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two. Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. Motivation and name Many devices indicate position by closing and opening switches. If that device uses natural binary codes, positions 3 and 4 are next to each other but all three bits of the binary representation differ: : The problem with natural binary codes is that physical switches are not ideal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Path
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs. Definition Let G be a group and S be a generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a color c_s. * For every g \in G and s \in S, there is a directed edge of color c_s from the vertex corresponding to g to the one corresponding to gs. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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