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Permutation Class
In the study of permutations and permutation patterns, a permutation class is a set C of permutations for which every pattern within a permutation in C is also in C. In other words, a permutation class is a hereditary property of permutations, or a downset in the permutation pattern order. A permutation class may also be known as a pattern class, closed class, or simply class of permutations. Every permutation class can be defined by the minimal permutations which do not lie inside it, its ''basis''., Definition 8.1.3, p. 318. A principal permutation class is a class whose basis consists of only a single permutation. Thus, for instance, the stack-sortable permutations form a principal permutation class, defined by the forbidden pattern 231. However, some other permutation classes have bases with more than one pattern or even with infinitely many patterns. A permutation class that does not include all permutations is called proper. In the late 1980s, Richard Stanley and Herbert Wilf ... [...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] |
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] |
Hereditary Property
In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of ''subobject'' depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory. In topology In topology, a topological property is said to be ''hereditary'' if whenever a topological space has that property, then so does every subspace of it. If the latter is true only for closed subspaces, then the property is called ''weakly hereditary'' or ''closed-hereditary''. For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary. Connectivity is not weakly hereditary. If ''P'' is a property of a topological space ''X'' and every subspace also has property ''P'', then ''X'' is said to be "hereditarily ''P''". In combinatorics and graph theory Hereditary properties occur throughout combinatorics a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (order Theory)
In mathematics, mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory. Definitions A subset of a partially ordered set (P, \leq) is an ideal, if the following conditions hold: # is non-empty, # for every ''x'' in and ''y'' in ''P'', implies that ''y'' is in ( is a lower set), # for every ''x'', ''y'' in , there is some element ''z'' in , such that and ( is a directed set). While this is the most general way to define an ideal for arbitrary posets, it was originally defined for Lattice (order), lattices only. In this case, the following equivalent definition can be given: a subset of a lattice (P, \leq) is an ideal if and only if it is a lower set that is closed under finite join and me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stack-sortable Permutation
In mathematics and computer science, a stack-sortable permutation (also called a tree permutation) is a permutation whose elements may be sorted by an algorithm whose internal storage is limited to a single stack data structure. The stack-sortable permutations are exactly the permutations that do not contain the permutation pattern 231; they are counted by the Catalan numbers, and may be placed in bijection with many other combinatorial objects with the same counting function including Dyck paths and binary trees. Sorting with a stack The problem of sorting an input sequence using a stack was first posed by , who gave the following linear time algorithm (closely related to algorithms for the later all nearest smaller values problem): *Initialize an empty stack *For each input value ''x'': **While the stack is nonempty and ''x'' is larger than the top item on the stack, pop the stack to the output **Push ''x'' onto the stack *While the stack is nonempty, pop it to the output Knuth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard P
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic language">Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", " Dick", " Dickon", " Dickie", " Rich", " Rick", "Rico (name), Rico", " Ricky", and more. Richard is a common English (the name was introduced into England by the Normans), German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Portuguese and Spanish "Ricardo" and the Italian "Riccardo" (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Anders ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Wilf
Herbert Saul Wilf (June 13, 1931 – January 7, 2012) was an American mathematician, specializing in combinatorics and graph theory. He was the Thomas A. Scott Professorship of Mathematics, Thomas A. Scott Professor of Mathematics in Combinatorial Analysis and Computing at the University of Pennsylvania. He wrote numerous books and research papers. Together with Neil Calkin he founded ''The Electronic Journal of Combinatorics'' in 1994 and was its editor-in-chief until 2001. Biography Wilf was the author of numerous papers and books, and was adviser and mentor to many students and colleagues. His collaborators include Doron Zeilberger and Donald Knuth. One of Wilf's former students is Richard Garfield, the creator of the collectible card game ''Magic: The Gathering''. He also served as a thesis advisor for E. Roy Weintraub in the late 1960s. Wilf died of a progressive neuromuscular disease in 2012. Awards In 1996, Wilf received the Deborah and Franklin Haimo Awards for Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . and other numbers ''x'' such that would be an upper bound for ''S''. The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanley–Wilf Conjecture
The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, states that the growth rate of every proper permutation class is Exponential growth, singly exponential. It was proved by and is no longer a conjecture. Marcus and Tardos actually proved a different conjecture, due to , which had been shown to imply the Stanley–Wilf conjecture by . Statement The Stanley–Wilf conjecture states that for every permutation ''β'', there is a constant ''C'' such that the number , ''S''''n''(''β''), of permutations of length ''n'' which avoid ''β'' as a permutation pattern is at most ''C''''n''. As observed, this is equivalent to the convergence of the Limit (mathematics), limit :\lim_ \sqrt[n]. The upper bound given by Marcus and Tardos for ''C'' is Exponential function, exponential in the length of ''β''. A stronger conjecture of had stated that one could take ''C'' to be , where ''k'' denotes the length of ''β'', but this con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adam Marcus (mathematician)
Adam Wade Marcus (born 1979) is an American mathematician. He held the Chair of Combinatorial Analysis in the Institute of Mathematics at the École Polytechnique Fédérale de Lausanne until February 2023. The team of Marcus, Daniel Spielman and Nikhil Srivastava was awarded the Pólya Prize in 2014 for their resolution of the Kadison–Singer problem and later the Michael and Sheila Held Prize in 2021 for their solution to long-standing conjectures in the study of Ramanujan graphs. History Marcus grew up in Marietta, Georgia and was a boarding student at the Darlington School in Rome, Georgia. He attended the Washington University in St. Louis for his undergraduate degree, where he was a Compton Fellow. He then completed his doctoral studies under the supervision of Prasad Tetali at the Georgia Institute of Technology. Following his graduation in 2008, he spent four years as a Gibbs Assistant Professor in Applied Mathematics at Yale University. In 2012, Marcus cofounded Crispl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gábor Tardos
Gábor Tardos (born 11 July 1964) is a Hungarian mathematician, currently a professor at Central European University and previously a Canada Research Chair at Simon Fraser University. He works mainly in combinatorics and computer science. He is the younger brother of Éva Tardos. Education and career Gábor Tardos received his PhD in Mathematics from Eötvös University, Budapest in 1988. His counsellors were László Babai and Péter Pálfy. He held postdoctoral posts at the University of Chicago, Rutgers University, University of Toronto and the Princeton Institute for Advanced Study. From 2005 to 2013, he served as a Canada Research Chair of discrete and computational geometry at Simon Fraser University. He then returned to Budapest to the Alfréd Rényi Institute of Mathematics where he has served as a research fellow since 1991. Mathematical results Tardos started with a result in universal algebra: he exhibited a maximal clone of order-preserving operations that is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilf Equivalence
In the study of permutations and permutation patterns, Wilf equivalence is an equivalence relation on permutation classes. Two permutation classes are Wilf equivalent when they have the same numbers of permutations of each possible length, or equivalently if they have the same 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 invo ...s. The equivalence classes for Wilf equivalence are called Wilf classes; they are the combinatorial classes of permutation classes. The counting functions and Wilf equivalences among many specific permutation classes are known. Wilf equivalence may also be described for individual permutations rather than permutation classes. In this context, two permutations are said to be Wilf equivalent if the principal permutation classes formed by forb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
