scholarly journals Adjacent transformations in permutations

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Adeline Pierrot ◽  
Dominique Rossin ◽  
Julian West
Keyword(s):  
Equivalence Class ◽  
Equivalence Classes ◽  
Set Of Permutations ◽  
International Audience ◽  
Different Order

International audience We continue a study of the equivalence class induced on $S_n$ when one is permitted to replace a consecutive set of elements in a permutation with the same elements in a different order. For each possible set of allowed replacements, we characterise and/or enumerate the set of permutations reachable from the identity. In some cases we also count the number of equivalence classes. Nous étudions dans cet article les classes d'équivalence sur les permutations obtenues en remplaçant un ensemble consécutif de valeurs par ces même valeurs mais dans un ordre différent. Nous étudions l'ensemble des remplacements possibles de longueur 3 et pour chacun d'entre eux caractérisons et énumérons les permutations de la classe de l'identité. Pour certains ensembles, nous calculons de même le nombre de classes d'équivalence.

scholarly journals On the finiteness of the derived equivalence classes of some stable endomorphism rings

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1157-1183 ◽  
Author(s):  
Jenny August
Keyword(s):  
Equivalence Class ◽  
Minimal Model ◽  
Equivalence Classes ◽  
Finite Dimensional Algebra ◽  
Derived Equivalence ◽  
Endomorphism Rings ◽  
Dimensional Algebra ◽  
Rigid Objects ◽  
Finite Dimensional ◽  
Frobenius Category

Abstract We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $$f :X \rightarrow {\mathrm{Spec}\;}\,R$$ f : X → Spec R , where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $$A_{{\text {con}}}$$ A con known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of $$A_{\text {con}}$$ A con and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f. This provides evidence towards a key conjecture in the area.

Regent Results in Comma-Free Codes

1963 ◽  
Vol 15 ◽  
pp. 178-187 ◽  
Author(s):  
B. H. Jiggs
Keyword(s):  
Equivalence Class ◽  
Cyclic Permutation ◽  
Equivalence Classes ◽  
The Other ◽  
Complete Class

A set D of k-letter words is called a comma-free dictionary (2), if whenever (a1a2 . . . ak) and (b1b2 . . . bk) are in D, the "overlaps" (a2a3 . . . akb1), (a3a4 . . . akb1b2), . . . , (akb1 . . . bk-1) are not in D. We say that two k-letter words are in the same equivalence class if one is a cyclic permutation of the other. An equivalence class is called complete if it contains k distinct members. Comma-freedom is violated if we choose words from incomplete equivalence classes, or if more than one word is chosen from the same complete class.

Scalable l-Diversity

2019 ◽  
Vol 14 (2) ◽  
pp. 27-40
Author(s):  
Udai Pratap Rao ◽  
Brijesh B. Mehta ◽  
Nikhil Kumar
Keyword(s):  
Equivalence Class ◽  
Information Loss ◽  
Equivalence Classes ◽  
Data Publishing ◽  
Programming Paradigm ◽  
Research Areas ◽  
Complete Dataset ◽  
Input Dataset ◽  
Privacy Preserving Data Publishing ◽  
Cost Penalty

Privacy preserving data publishing is one of the most demanding research areas in the recent few years. There are more than billions of devices capable to collect the data from various sources. To preserve the privacy while publishing data, algorithms for equivalence class generation and scalable anonymization with k-anonymity and l-diversity using MapReduce programming paradigm are proposed in this article. Equivalence class generation algorithms divide the datasets into equivalence classes for Scalable k-Anonymity (SKA) and Scalable l-Diversity (SLD) separately. These equivalence classes are finally fed to the anonymization algorithm that calculates the Gross Cost Penalty (GCP) for the complete dataset. The value of GCP gives information loss in input dataset after anonymization.

scholarly journals Bipolar Fuzzy Relations

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1044 ◽  
Author(s):  
Jeong-Gon Lee ◽  
Kul Hur
Keyword(s):  
Equivalence Class ◽  
Level Set ◽  
Equivalence Relation ◽  
Level Sets ◽  
Equivalence Classes ◽  
Fuzzy Relation ◽  
Fuzzy Relations ◽  
Transitive Relation ◽  
Fuzzy Partition

We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an ( a , b ) -level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their ( a , b ) -level sets.

scholarly journals The binomial equivalence classes of finite words

2020 ◽  
Vol 30 (07) ◽  
pp. 1375-1397
Author(s):  
Marie Lejeune ◽  
Michel Rigo ◽  
Matthieu Rosenfeld
Keyword(s):  
Nilpotent Group ◽  
Equivalence Class ◽  
Free Monoid ◽  
Equivalence Classes ◽  
Single Element ◽  
Free Nilpotent Group ◽  
Context Free ◽  
Abelian Equivalence

Two finite words [Formula: see text] and [Formula: see text] are [Formula: see text]-binomially equivalent if, for each word [Formula: see text] of length at most [Formula: see text], [Formula: see text] appears the same number of times as a subsequence (i.e., as a scattered subword) of both [Formula: see text] and [Formula: see text]. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the [Formula: see text]-binomial equivalence. We provide an algorithm generating the [Formula: see text]-binomial equivalence class of a word. For [Formula: see text] and alphabet of [Formula: see text] or more symbols, the language made of lexicographically least elements of every [Formula: see text]-binomial equivalence class and the language of singletons, i.e., the words whose [Formula: see text]-binomial equivalence class is restricted to a single element, are shown to be non-context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-[Formula: see text] group (also called free nilpotent group of class [Formula: see text]) on [Formula: see text] generators is isomorphic to the quotient of the free monoid [Formula: see text] by the [Formula: see text]-binomial equivalence.

Transfer of Social Attributions in Stimulus Equivalence Classes by Preschool Children

1997 ◽  
Vol 80 (1) ◽  
pp. 3-21 ◽  
Author(s):  
Mark Egli ◽  
Beth Joseph ◽  
Travis Thompson
Keyword(s):  
Preschool Children ◽  
Antisocial Behavior ◽  
Equivalence Class ◽  
Stimulus Equivalence ◽  
Computer Game ◽  
Equivalence Classes ◽  
Antisocial Behaviors ◽  
Conditional Discriminations ◽  
Response Options ◽  
Social Attributions

The transfer of social attributions within stimulus-equivalence classes comprised of photographs of children was examined. Five children (mean age: 4 yr., 2 mo.) were taught conditional discriminations sufficient for the emergence of two 3-member equivalence classes (A1-B1-C1 and A2-B2-C2). Social attributions were established by using two photographs to identify fictional children who could facilitate (B1) or prevent (B2) the participant's reinforcement on a computer game. Transfer of attribution was assessed by asking the participants questions regarding predicted social behaviors by children in all six photographs. One set of questions pertained explicitly to the response-options of the computer game; a second set referred to other prosocial and antisocial behaviors. Three children chose photographs in response to questions consistent with their experience with members B1 and B2 of the shared equivalence class when the questions pertained to the computer game. One subject also selected photographs in response to questions about predicted prosocial and antisocial behavior which reflected her experience with the B1 and B2 photographs.

scholarly journals Mixed volumes of hypersimplices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Nous Montrons ◽  
Set Of Permutations ◽  
International Audience

International audience In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers. Dans ce résumé étendu nous considérons les volumes mixtes de combinaisons d’hyper-simplexes. Ces nombres, appelés les nombres Eulériens mixtes, ont été pour la première fois étudiés par A. Postnikov, et il a été montré qu’ils satisfont à de nombreuses propriétés reliées aux nombres Eulériens, au nombres de Catalan, aux coefficients binomiaux, etc. Nous donnons une interprétation combinatoire générale des nombres Eulériens mixtes, et nous prouvons combinatoirement les propriétés mentionnées ci-dessus. En particulier, nous montrons que chaque nombre Eulérien mixte compte les éléments d’un certain sous-ensemble de l’ensemble des permutations $S_n$. Nous établissons également plusieurs nouvelles propriétés des nombres Eulériens mixtes grâce à notre méthode. Pour finir, nous introduisons une généralisation en type $B$ des nombres Eulériens mixtes, et nous en donnons une interprétation combinatoire analogue.

scholarly journals Permutations Containing and Avoiding $\textit{123}$ and $\textit{132}$ Patterns

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Aaron Robertson
Keyword(s):  
Set Of Permutations ◽  
International Audience

International audience We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern, equals $(n-2)2^{n-3}$, for $n \ge 3$. We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is $(n-3)(n-4)2^{n-5}$, for $n \ge 5$.

scholarly journals Universal cycles for permutation classes

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Michael Albert ◽  
Julian West
Keyword(s):  
General Result ◽  
Universal Cycles ◽  
Set Of Permutations ◽  
International Audience

International audience We define a universal cycle for a class of $n$-permutations as a cyclic word in which each element of the class occurs exactly once as an $n$-factor. We give a general result for cyclically closed classes, and then survey the situation when the class is defined as the avoidance class of a set of permutations of length $3$, or of a set of permutations of mixed lengths $3$ and $4$. Nous définissons un cycle universel pour une classe de $n$-permutations comme un mot cyclique dans lequel chaque élément de la classe apparaît une unique fois comme $n$-facteur. Nous donnons un résultat général pour les classes cycliquement closes, et détaillons la situation où la classe de permutations est définie par motifs exclus, avec des motifs de taille $3$, ou bien à la fois des motifs de taille $3$ et de taille $4$.

scholarly journals Some simple varieties of trees arising in permutation analysis

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Mathilde Bouvel ◽  
Marni Mishna ◽  
Cyril Nicaud
Keyword(s):  
Related Species ◽  
Asymptotic Formulas ◽  
Genome Comparison ◽  
Set Of Permutations ◽  
International Audience ◽  
Interval Trees ◽  
Number Of Leaves ◽  
Permutation Analysis

International audience After extending classical results on simple varieties of trees to trees counted by their number of leaves, we describe a filtration of the set of permutations based on their strong interval trees. For each subclass we provide asymptotic formulas for number of trees (by leaves), average number of nodes of fixed arity, average subtree size sum, and average number of internal nodes. The filtration is motivated by genome comparison of related species. Nous commençons par étendre les résultats classiques sur les variétés simples d'arbres aux arbres comptés selon leur nombre de feuilles, puis nous décrivons une filtration de l'ensemble des permutations qui repose sur leurs arbres des intervalles communs. Pour toute sous-classe, nous donnons des formules asymptotiques pour le nombre d'arbres (comptés selon les feuilles), le nombre moyen de nœuds d'arité fixée, la moyenne de la somme des tailles des sous-arbres, et le nombre moyen de nœuds internes. Cette filtration est motivée par des problématiques de comparaison de génomes.

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