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(p,q) Shuffle
In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of n items that can be obtained by a single riffle shuffle, in which a sorted deck of n cards is cut into two packets and then the two packets are interleaved (e.g. by moving cards one at a time from the bottom of one or the other of the packets to the top of the sorted deck). Beginning with an ordered set (1 rising sequence), mathematically a riffle shuffle is defined as a permutation on this set containing 1 or 2 rising sequences. The permutations with 1 rising sequence are the identity permutations. As a special case of this, a (p,q)-shuffle, for numbers p and q with p+q=n, is a riffle in which the first packet has p cards and the second packet has q cards.Weibel, Charles (1994). ''An Introduction to Homological Algebra'', p. 181. Cambridge University Press, Cambridge. Combinatorial enumeration Since a (p,q)-shuffle is completely determine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Vexillary Permutation
In mathematics, a vexillary permutation is a permutation ''μ'' of the positive integers containing no subpermutation isomorphic to the permutation (2143); in other words, there do not exist four numbers ''i'' < ''j'' < ''k'' < ''l'' with ''μ''(''j'') < ''μ''(''i'') < ''μ''(''l'') < ''μ''(''k''). They were introduced by . The word "vexillary" means flag-like, and comes from the fact that vexillary permutations are related to flags
A flag is a piece of fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and flags have ... of [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gilbreath Shuffle
A Gilbreath shuffle is a way to shuffle a deck of cards, named after mathematician Norman Gilbreath (also known for Gilbreath's conjecture). Gilbreath's principle describes the properties of a deck that are preserved by this type of shuffle, and a Gilbreath permutation is a permutation that can be formed by a Gilbreath shuffle.. Description A Gilbreath shuffle consists of the following two steps: *Deal off any number of the cards from the top of a deck to form a second pile of cards. *Riffle the new pile with the remainder of the deck. It differs from the more commonly used procedure of cutting a deck into two piles and then riffling the piles, in that the first step of dealing off cards reverses the order of the cards in the new pile, whereas cutting the deck would preserve this order. Gilbreath's principle Although seemingly highly random, Gilbreath shuffles preserve many properties of the initial deck. For instance, if the initial deck of cards alternates between black and red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Exterior Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade (geometry), blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude (mathematics), magnitude of a bivector, -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the Parallelepiped#Parallelotope, parallelotope defined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hopf Algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Shuffle Algebra
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product ''X'' ⧢ ''Y'' of two words ''X'', ''Y'': the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation. The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set. Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words. The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative. Shuffle product The shuffle product of words of lengths ''m'' and ''n'' is a sum over the ways of interleaving the two words, as shown in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Out Shuffle
The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling playing cards, in which half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to the table interleaved. Diaconis, Graham, and Kantor also call this the technique, when used in magic. Mathematicians use the term "faro shuffle" to describe a precise rearrangement of a deck into two equal piles of 26 cards which are then interleaved perfectly. Description A right-handed practitioner holds the cards from above in the left hand and from below in the right hand. The deck is separated into two preferably equal parts by simply lifting up half the cards with the right thumb slightly and pushing the left hand's packet forward away from the right hand. The two packets are often crossed and tapped against each other to align them. They are then pushed together on the short sides and bent either up or down. The cards will then alt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
In Shuffle
The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling playing cards, in which half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to the table interleaved. Diaconis, Graham, and Kantor also call this the technique, when used in magic. Mathematicians use the term "faro shuffle" to describe a precise rearrangement of a deck into two equal piles of 26 cards which are then interleaved perfectly. Description A right-handed practitioner holds the cards from above in the left hand and from below in the right hand. The deck is separated into two preferably equal parts by simply lifting up half the cards with the right thumb slightly and pushing the left hand's packet forward away from the right hand. The two packets are often crossed and tapped against each other to align them. They are then pushed together on the short sides and bent either up or down. The cards will then alter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Permutation Pattern
In combinatorial mathematics and theoretical computer science, a (classical) permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation as a sequence of entries representing the result of applying the permutation to the sequence 123...; for instance the sequence 213 represents the permutation on three elements that swaps elements 1 and 2. If π and σ are two permutations represented in this way (these variable names are standard for permutations and are unrelated to the number pi), then π is said to ''contain'' σ as a ''pattern'' if some subsequence of the entries of π has the same relative order as all of the entries of σ. For instance, permutation π contains the pattern 213 whenever π has three entries ''x'', ''y'', and ''z'' that appear within π in the order ''x''...''y''...''z'' but whose values are ordered as ''y'' < ''x'' < ''z'', the same as the ordering of the values in the permutation 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Shuffling
Shuffling is a technique used to randomize a deck of playing cards, introducing an element of chance into card games. Various shuffling methods exist, each with its own characteristics and potential for manipulation. One of the simplest shuffling techniques is the overhand shuffle, where small packets of cards are transferred from one hand to the other. This method is easy to perform but can be manipulated to control the order of cards. Another common technique is the riffle shuffle, where the deck is split into two halves and interleaved. This method is more complex but minimizes the risk of exposing cards. The Gilbert–Shannon–Reeds model suggests that seven riffle shuffles are sufficient to thoroughly randomize a deck, although some studies indicate that six shuffles may be enough. Other shuffling methods include the Hindu shuffle, commonly used in Asia, and the pile shuffle, where cards are dealt into piles and then stacked. The Mongean shuffle involves a specific seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
New York Times
''The New York Times'' (''NYT'') is an American daily newspaper based in New York City. ''The New York Times'' covers domestic, national, and international news, and publishes opinion pieces, investigative reports, and reviews. As one of the longest-running newspapers in the United States, the ''Times'' serves as one of the country's Newspaper of record, newspapers of record. , ''The New York Times'' had 9.13 million total and 8.83 million online subscribers, both by significant margins the List of newspapers in the United States, highest numbers for any newspaper in the United States; the total also included 296,330 print subscribers, making the ''Times'' the second-largest newspaper by print circulation in the United States, following ''The Wall Street Journal'', also based in New York City. ''The New York Times'' is published by the New York Times Company; since 1896, the company has been chaired by the Ochs-Sulzberger family, whose current chairman and the paper's publ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Probability Distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical description of a Randomness, random phenomenon in terms of its sample space and the Probability, probabilities of Event (probability theory), events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that fair coin, the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. Introduction A prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
