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Subpermutation of a longer permutation
theoretical computer science, a (classical) permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation
Permutation_pattern
Mathematical version of an order change
Levi-Civita symbol List of permutation topics Major index Permutation category Permutation group Permutation pattern Permutation representation (symmetric
Permutation
Ordering obtained by a single shuffle
In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is a permutation of a set of n {\displaystyle
Riffle_shuffle_permutation
data structure. The stack-sortable permutations are exactly the permutations that do not contain the permutation pattern 231; they are counted by the Catalan
Stack-sortable_permutation
the forbidden permutation patterns 2413 and 3142; they are also the permutations whose permutation graphs are cographs and the permutations that realize
Separable_permutation
In the study of permutation patterns, there has been considerable interest in enumerating specific permutation classes, especially those with relatively
Enumerations of specific permutation classes
Enumerations_of_specific_permutation_classes
permutations that do not contain the permutation patterns 231 or 312. That is, no three elements in the permutation (regardless of whether they are consecutive)
Layered_permutation
study of permutations and permutation patterns, a superpattern or universal permutation is a permutation that contains all of the patterns of a given
Superpattern
permutations and permutation patterns, a permutation class is a set C {\displaystyle C} of permutations for which every pattern within a permutation in
Permutation_class
Method of shuffling a deck of cards
Equivalently, in terms of permutation patterns, the Gilbreath permutations are the permutations that avoid the two patterns 132 and 312. A Gilbreath shuffle
Gilbreath_shuffle
Selection in a particular order
In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of
Partial_permutation
In the theory of permutation patterns, a skew-merged permutation is a permutation that can be partitioned into an increasing sequence and a decreasing
Skew-merged_permutation
Permutation graph Permutation pattern Permutation polynomial Permutohedron Rencontres numbers Robinson–Schensted correspondence Sum of permutations:
List_of_permutation_topics
mathematics, a Baxter permutation is a permutation σ ∈ S n {\displaystyle \sigma \in S_{n}} which satisfies the following generalized pattern avoidance property:
Baxter_permutation
sum of permutations are two operations to combine shorter permutations into longer ones. Given a permutation π of length m and the permutation σ of length
Skew and direct sums of permutations
Skew_and_direct_sums_of_permutations
empty set. In the study of permutation patterns, a combinatorial class of permutation classes, enumerated by permutation length, is called a Wilf class
Combinatorial_class
Number line and triangular tiling's symmetry mathematical structure
group S n {\displaystyle S_{n}} , a permutation is fully commutative if and only if it avoids the permutation pattern 321, that is, if and only if its one-line
Affine_symmetric_group
Sufficiently long sequences of numbers have long monotonic subsequences
language of permutation patterns as stating that every permutation of length at least (r − 1)(s − 1) + 1 must contain the pattern 12⋯r or the pattern s⋯21.
Erdős–Szekeres_theorem
String in combinatorial math
= 7 is still 5884. Superpattern, a permutation that contains each permutation of n symbols as a permutation pattern De Bruijn sequence, a similar problem
Superpermutation
Theorem that the growth rate of every proper permutation class is singly exponential
for every permutation β, there is a constant C such that the number |Sn(β)| of permutations of length n which avoid β as a permutation pattern is at most
Stanley–Wilf_conjecture
Type of permutation in combinatorial mathematics
In combinatorial mathematics, a Stirling permutation of order k is a permutation of the multiset 1, 1, 2, 2, ..., k, k (with two copies of each value
Stirling_permutation
Czech mathematician (born 1966)
in Prague. Klazar is known for his work on pattern avoidance in discrete structures (such as permutations and set partitions) and on extremal problems
Martin_Klazar
Canadian mathematician
permutations and pattern restricted permutations" (2005), demonstrated the use of the substitution decomposition in the context of permutation patterns, providing
Michael_H._Albert
Matrix with one nonzero entry in each row and column
mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly
Generalized permutation matrix
Generalized_permutation_matrix
the study of permutations and permutation patterns, Wilf equivalence is an equivalence relation on permutation classes. Two permutation classes are Wilf
Wilf_equivalence
American mathematician (born 1969)
Circle, and an expert in the combinatorial enumeration of permutations with forbidden patterns. Stankova was born in Ruse, Bulgaria (then the People's Republic
Zvezdelina_Stankova
Icelandic mathematician
lies in enumerative combinatorics, especially the study of permutation patterns and permutation statistics. He is a research professor (emeritus) in the
Einar_Steingrímsson
Sensor that can create 3D scans using visible light
Changsoo; Lee, Sang Wook; Park, Rae-Hong (2012). "Colour-stripe permutation pattern for rapid structured-light range imaging". Optics Communications
Structured-light_3D_scanner
Type of permutation
mathematics, a vexillary permutation is a permutation μ of the positive integers containing no subpermutation isomorphic to the permutation (2143); in other words
Vexillary_permutation
Natural number
Aronson's sequence. At most 226 different permutation patterns can occur within a single 9-element permutation. Sloane, N. J. A. (ed.). "Sequence A007770
226_(number)
English mathematician
mathematician and computer scientist known for his work in the theory of permutation patterns and for contributions to algorithm design, data structures, and algebra
Michael_D._Atkinson
Recursive integer sequence
Cn is the number of permutations of {1, ..., n} that avoid the permutation pattern 123 (or, alternatively, any of the other patterns of length 3); that
Catalan_number
American mathematician
mathematics at DePaul University in Chicago. Her research focuses on permutation patterns, and has also included work in algebraic combinatorics, discrete
Bridget_Tenner
Israeli mathematician (born 1968)
its applications. In particular, he is interested in permutation patterns, colored permutations, set partitions, combinatorics on words, and compositions
Toufik_Mansour
Russian-British mathematician
is best known for his book Patterns in permutations and words (2011), an introduction to the field of permutation patterns. He is also the author (with
Sergey_Kitaev
Equivalence of partially ordered sets
are called order types. Permutation pattern, a permutation that is order-isomorphic to a subsequence of another permutation Bloch (2011); Ciesielski
Order_isomorphism
Extraction of 3D data from digital images
Changsoo; Lee, Sang Wook; Park, Rae-Hong (2012). "Colour-stripe permutation pattern for rapid structured-light range imaging". Optics Communications
Computer_stereo_vision
Method of generating all permutations of n objects
possible permutations of n objects. It was first proposed by B. R. Heap in 1963. The algorithm minimizes movement: it generates each permutation from the
Heap's_algorithm
invented here or (NIH) syndrome Premature optimization Programming by permutation (or "programming by accident", or "programming by coincidence") Reinventing
List of software anti-patterns
List_of_software_anti-patterns
Count of the possible partitions of a set
dash, these permutations can be described as the permutations that avoid the pattern 1-23. The permutations that avoid the generalized patterns 12-3, 32-1
Bell_number
Chinese-style pattern used on pottery
pattern became the most popular and persistent of them, and in various permutations has remained in production to the present day. Characteristically the
Willow_pattern
Machine learning technique
by the rows of V {\displaystyle V} . To understand the permutation invariance and permutation equivariance properties of QKV attention, let A ∈ R m ×
Attention_(machine_learning)
American mathematician (born 1971)
titled Permutation Patterns and Continued Fractions. In the paper, they "find a generating function for the number of (132)-avoiding permutations that have
Aaron Robertson (mathematician)
Aaron_Robertson_(mathematician)
Block cipher
achieve quick diffusion. The permutation step swaps the positions of the words according to a constant pattern. Bit-level permutation is not achieved in this
Threefish
Graph layout on multiple half-planes
if and only if its initial order is described by a permutation that avoids the permutation pattern 231. Since then, there has been much work on similar
Book_embedding
English mathematician
enumeration of grid classes of permutations and for his work on enumerating the class of permutations avoiding the pattern 1324. He is also known for devising
David_Bevan_(mathematician)
Romanian-American mathematician
research concerned combinatorics: she was a pioneer in the study of permutation patterns, and an expert on noncrossing partitions. Simion was one of the top
Rodica_Simion
Set of cryptographic hash functions
sponge construction. The sponge construction is based on a pseudorandom permutation, and allows inputting ("absorbing" in sponge terminology) any amount
SHA-3
Method of encryption
In cryptography, a transposition cipher (also known as a permutation cipher) is a method of encryption which scrambles the positions of characters (transposition)
Transposition_cipher
Method in speedcubing
Cross, First Two Layers (F2L), Orientation of the Last Layer (OLL), and Permutation of the Last Layer (PLL). Alongside the Roux and ZZ methods, it is regarded
CFOP_method
Combinatorial problem
scaled converges to a normal distribution. Alternating permutation Permutation pattern and pattern avoidance Counting local maxima and/or local minima in
Longest alternating subsequence
Longest_alternating_subsequence
Tree-based ensemble machine learning methods
estimate of the generalization error. Measuring variable importance through permutation. The report also offers the first theoretical result for random forests
Random_forest
Canadian computer scientist
include proving the NP-completeness of finding permutation patterns, and of finding derangements in permutation groups. Lubiw was named an ACM Distinguished
Anna_Lubiw
American mathematician
Combinatorics (2006), the British Combinatorial Conference (2011), and Permutation Patterns (2015). He has graduated 15 Ph.D. students. During his time at Michigan
Bruce_Sagan
Rodica Simion (1955–2000), Romanian-American pioneer in the study of permutation patterns Valeria Simoncini (born 1966), Italian numerical analyst Lao Genevra
List_of_women_in_mathematics
ball graphs in higher dimensions. The permutation graphs coming from permutations with a forbidden permutation pattern have bounded twin-width. This allows
Twin-width
Pattern that has no predecessors
known examples of groups that are not surjunctive. In Greg Egan's novel Permutation City, the protagonist uses a Garden of Eden configuration to create a
Garden of Eden (cellular automaton)
Garden_of_Eden_(cellular_automaton)
Combinatorial algorithm
F. Trotter that generates all of the permutations of n {\displaystyle n} elements. Each two adjacent permutations in the resulting sequence differ by swapping
Steinhaus–Johnson–Trotter algorithm
Steinhaus–Johnson–Trotter_algorithm
Mathematical counting-out question
computer science and mathematics, the Josephus problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game. Such
Josephus_problem
of class-invariance found in pattern recognition is permutation-invariance, i.e. invariance of the class to a permutation of elements in a structured input
Prior knowledge for pattern recognition
Prior_knowledge_for_pattern_recognition
Branch of mathematics that studies the properties of groups
Lie group, are used for pattern recognition and other image processing techniques. In combinatorics, the notion of permutation group and the concept of
Group_theory
Integer sequence
individual elements of decimal sequences immediately settle into a permutation of the form a0 b1 c2 d3 e4 f5 g6 h7 i8 j9 where here the letters a–j
Look-and-say_sequence
Numeral system in combinatorics
as factoradic), is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function
Factorial_number_system
3D combination puzzle
preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If the permutations reached through disassembly
Rubik's_Cube
Study of health and disease within a population
is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population
Epidemiology
Sounding continually changing mathematical permutations on bells
at each change. In method ringing, the ringers are guided from permutation to permutation by following the rules of a method. Ringers typically learn a
Method_ringing
4×4×4 Rubik's cube variation
colours. An odd permutation of the corners implies an odd permutation of the centres and vice versa; however, even and odd permutations of the centres
Rubik's_Revenge
Type of composite number with an even number of digits
({a_{k}}{a_{k-1}}...{a_{2}}{a_{1}}{b_{k}}{b_{k-1}}...{b_{2}}{b_{1}})} are a permutation of the 2 k {\displaystyle 2k} digits of N {\displaystyle N} . The two
Vampire_number
Type of cipher
text. For each key K, EK is a permutation (a bijective mapping) over the set of input blocks. Each key selects one permutation from the set of ( 2 n ) ! {\displaystyle
Block_cipher
Discrete mathematics decomposition
rectangulations and permutation classes. In the following table p 1 {\displaystyle p_{1}} and p 2 {\displaystyle p_{2}} refer to the mesh patterns given by p 1
Rectangulations
Game playing card
The effort is purported to have driven Leffler insane. Manual random permutation is an onerous and time-consuming task that limited the number of Bingo
Bingo_card
3D symmetry group
symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the
Octahedral_symmetry
Number of ways to pair up n objects
Every pattern of pairwise connections between n people defines an involution, a permutation of the people that is its own inverse. In this permutation, each
Telephone number (mathematics)
Telephone_number_(mathematics)
Cycle through all length-k sequences
while preserving order. This process defines the Standard Permutation. Write this permutation in cycle notation with the smallest position in each cycle
De_Bruijn_sequence
Number in combinatorics
first occurrences of each number in sorted order) that avoid the permutation patterns 12312 and 121323. The closely related large Schröder numbers are
Schröder–Hipparchus_number
Permanent Permutation Enumerations of specific permutation classes Josephus permutation Permutation matrix Permutation pattern Permutation (disambiguation)
Index of combinatorics articles
Index_of_combinatorics_articles
Algorithms to complete a sudoku
partial permutation of N. Let T = { X : X is a row, column, or block of Q }, so T has 27 elements. An arrangement is either a partial permutation or a permutation
Sudoku_solving_algorithms
Mathematical abelian group
group of permutations of these three elements, that is, the symmetric group S 3 {\displaystyle S_{3}} . The Klein four-group's permutations of its own
Klein_four-group
Non-commutative group with 6 elements
multiple of one-third of a full turn (0°, 120°, 240°).. Every possible permutation of the triangle's vertices under the dihedral group D3 constitutes such
Dihedral_group_of_order_6
mostly due to safety concerns. The 71 km pattern of Circuito delle Madonie is still used in the same permutation as the original race, and as such the Targa
Targa_Florio_Rally
Geometry with 7 points and 7 lines
permutation 21 permutations with two 2-cycles 42 permutations with a 4-cycle and a 2-cycle 56 permutations with two 3-cycles The 48 permutations with a complete
Fano_plane
Number that when multiplied by another number moves its last digit to its front
Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related Cyclic Digit-reassembly Parasitic Primeval Transposable Divisor-related
Parasitic_number
Competitive solving of combination puzzles
PLL (Permutation of the Last Layer) algorithms. OLL and PLL use 57 and 21 algorithms, respectively. Cross, First 2 Layers, Orientation, Permutation (CFOP)
Speedcubing
Annual entrance test held in India
Functions Complex Numbers and Quadratic Equations Linear Inequalities Permutations and Combinations Binomial Theorem Sequences and Series Straight Lines
IISER_Aptitude_Test
Number that permute or shift cyclically when multiplied by another number
corresponding fractions. The greatest common divisor (gcd) between any cyclic permutation of an m-digit integer and 10m − 1 is constant. Expressed as a formula
Transposable_integer
Property of objects inherited by all their subobjects
names. For example, in the context of permutation patterns, hereditary properties are typically called permutation classes. In graph theory, a hereditary
Hereditary_property
Puzzle game involving sliding pieces
Klotski puzzle An unsolvable puzzle due to the pieces not being in an even permutation Fifteen puzzle Klotski Minus Cube Rush Hour Sokoban Rubik's Slide Ro
Sliding_puzzle
Larger variant of the Rubik's cube
ways to arrange the central edges, since an odd permutation of the corners implies an odd permutation of these pieces as well. There are 211 ways that
V-Cube_7
8×8×8 version of Rubik's Cube
8 corners, 72 edges, and 216 centers. Any permutation of the corners is possible, including odd permutations. Seven of the corners can be independently
V-Cube_8
Contrapuntal musical form based on a subject that recurs in imitation
not purely a permutation fugue, as it does have episodes between permutation expositions. Invertible counterpoint is essential to permutation fugues but
Fugue
In mathematics, invariant of square matrices
corresponding permutation (which is + 1 {\displaystyle +1} for an even number of permutations and is − 1 {\displaystyle -1} for an odd number of permutations). Once
Determinant
Describes statistically the splitting of primes in a given Galois extension of Q
K / Q ) {\displaystyle G=\operatorname {Gal} (K/\mathbb {Q} )} is a permutation of the roots of P ( t ) {\displaystyle P(t)} in K {\displaystyle K} ;
Chebotarev_density_theorem
Points usable to draw any planar graph
but are instead constructed from superpatterns (permutations that contain all permutation patterns of a given size); the universal point sets constructed
Universal_point_set
Properties of the operation of a secure cipher
columns of the cipher. In substitution–permutation networks, diffusion is provided by permutation boxes (a.k.a. permutation layer). In the beginning of the 21st
Confusion_and_diffusion
Unconditionally convergent series converge absolutely
numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged
Riemann_series_theorem
Decryption of World War II cipher
Rejewski at the Polish General Staff's Cipher Bureau, using mathematical permutation group theory combined with French-supplied intelligence material obtained
Cryptanalysis_of_the_Enigma
Integer whose multiples are digit rotations
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely-known cyclic
Cyclic_number
Rubik's Cube configuration
Rubik's Cube, in which all the edge and corner pieces are in the correct permutation, and the eight corners are correctly oriented, but all twelve edges are
Superflip
Group of transformations under which the object is invariant
and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern
Symmetry_group
Algorithmic technique using hashing
been significant work on finding a family of permutations that is "min-wise independent" — a permutation family for which each element of the domain has
Locality-sensitive_hashing
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