scholarly journals Extending from bijections between marked occurrences of patterns to all occurrences of patterns

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Jeffrey Remmel ◽  
Mark Tiefenbruck
Keyword(s):  
Nous Avons ◽  
Large Class ◽  
Permutation Patterns ◽  
Open Problems ◽  
Combinatorial Objects ◽  
International Audience ◽  
General Method

International audience We consider two recent open problems stating that certain statistics on various sets of combinatorial objects are equidistributed. The first, posed by Anders Claesson and Svante Linusson, relates nestings in matchings on $\{1,2,\ldots,2n\}$ to occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We develop a general method that solves both of these problems and many more. We further employ the Garsia-Milne involution principle to obtain purely bijective proofs of these results. Nous considérons deux derniers problèmes ouverts indiquant que certaines statistiques sur les divers ensembles d'objets combinatoires sont équiréparties. La première, posée par Anders Claesson et Svante Linusson, concerne les imbrications dans des filtrages sur $\{1,2,\ldots,2n\}$ pour les occurrences d'un certain modèle de permutations dans $S_n$. La seconde, posée par Miles Jones et Jeffrey Remmel, concerne les occurrences d'une large classe de schémas de permutation consécutive aux évènements du même modèle dans les cycles de permutations. Nous développons une méthode générale qui résout ces deux problèmes et beaucoup plus. Nous avons également utiliser le principe d'involution Garsia-Milne pour obtenir des preuves purement bijectives de ces résultats.

Type Inference: Some Results, Some Problems1

1993 ◽  
Vol 19 (1-2) ◽  
pp. 87-125
Author(s):  
Paola Giannini ◽  
Furio Honsell ◽  
Simona Ronchi Della Rocca
Keyword(s):  
Large Class ◽  
Higher Order ◽  
Type Inference ◽  
Principal Type ◽  
Open Problems ◽  
Inference Problem ◽  
Type Assignment ◽  
Unification Problem

In this paper we investigate the type inference problem for a large class of type assignment systems for the λ-calculus. This is the problem of determining if a term has a type in a given system. We discuss, in particular, a collection of type assignment systems which correspond to the typed systems of Barendregt’s “cube”. Type dependencies being shown redundant, we focus on the strongest of all, Fω, the type assignment version of the system Fω of Girard. In order to manipulate uniformly type inferences we give a syntax directed presentation of Fω and introduce the notions of scheme and of principal type scheme. Making essential use of them, we succeed in reducing the type inference problem for Fω to a restriction of the higher order semi-unification problem and in showing that the conditional type inference problem for Fω is undecidable. Throughout the paper we call attention to open problems and formulate some conjectures.

scholarly journals Instrumentation des activités des tuteurs à l’aide d’un système multi-agents d’analyse automatique des interactions

2011 ◽  
Vol Volume 14 - 2011 - Special... ◽  
Author(s):  
Ilham Oumaira ◽  
Rochdi Messoussi ◽  
Raja TOUAHNI
Keyword(s):  
Nous Avons ◽  
Social Cognitive ◽  
Adaptive System ◽  
Interaction Analysis ◽  
Interaction Data ◽  
Multi Agent System ◽  
Système Adaptatif ◽  
International Audience ◽  
Multi Agent ◽  
Multi Agents

International audience Research presented in this article is dedicated to the tutor instrumentation in distance collaborative learning situations. We are particularly interested in the reuse of interaction analysis indicators. In this paper, we present our system SYSAT; a multi-agent system for monitoring the activities of learners. The aim of SYSAT is to reuse indicators (social, cognitive, emotional ...) reported in the literature, in an open and adaptive system. We tested our system on the interaction data from two experiments conducted with two master students of the Ibn Tofail University. The article presents the results and discusses the prospects for Research. Ce travail s'inscrit dans le cadre des recherches sur les Environnements Informatiques pour l'Apprentissage Humain (EIAH), et plus particulièrement dans l’assistance du tuteur dans le suivi des apprenants lors des activités d’apprentissage collaboratives en ligne. Cet article décrit l’architecture du système SYSAT, un système multi-agents d’analyse automatique des interactions. L’objectif de SYSAT est de réutiliser les indicateurs (sociaux, cognitifs, affectifs…) rapportés dans la littérature, au sein d’un système adaptatif et ouvert. Nous avons testé notre système sur les données d’interactions issues de deux expérimentations menées avec les étudiants de deux masters à l’université Ibn Tofail. L’article présente les résultats obtenus et évoque les perspectives de recherche.

scholarly journals The Limiting Distribution for the Number of Symbol Comparisons Used by QuickSort is Nondegenerate (Extended Abstract)

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Patrick Bindjeme ◽  
james Allen fill
Keyword(s):  
Large Class ◽  
Continuous Time ◽  
Random Variable ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean

International audience In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.

scholarly journals Formal Group Laws and Chromatic Symmetric Functions of Hypergraphs

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Power Series ◽  
Symmetric Functions ◽  
Formal Group ◽  
Combinatorial Interpretation ◽  
Recursive Structure ◽  
Combinatorial Objects ◽  
Formal Group Law ◽  
International Audience ◽  
Formal Group Laws ◽  
Group Law

International audience If $f(x)$ is an invertible power series we may form the symmetric function $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ which is called a formal group law. We give a number of examples of power series $f(x)$ that are ordinary generating functions for combinatorial objects with a recursive structure, each of which is associated with a certain hypergraph. In each case, we show that the corresponding formal group law is the sum of the chromatic symmetric functions of these hypergraphs by finding a combinatorial interpretation for $f^{-1}(x)$. We conjecture that the chromatic symmetric functions arising in this way are Schur-positive. Si $f(x)$ est une série entière inversible, nous pouvons former la fonction symétrique $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ que nous appelons une loi de groupe formel. Nous donnons plusieurs exemples de séries entières $f(x)$ qui sont séries génératrices ordinaires pour des objets combinatoires avec une structure récursive, chacune desquelles est associée à un certain hypergraphe. Dans chaque cas, nous donnons une interprétation combinatoire à $f^{-1}(x)$, ce qui nous permet de montrer que la loi de groupe formel correspondante est la somme des fonctions symétriques chromatiques de ces hypergraphes. Nous conjecturons que les fonctions symétriques chromatiques apparaissant de cette manière sont Schur-positives.

scholarly journals Tiling the Line with Triples

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Aaron Meyerowitz
Keyword(s):  
Greedy Algorithm ◽  
Direct Proof ◽  
Open Problems ◽  
One Dimensional ◽  
Proof By Contradiction ◽  
International Audience ◽  
The Subject ◽  
The One ◽  
Finite Digraph ◽  
Elegant Proof

International audience It is known the one dimensional prototile $0,a,a+b$ and its reflection $0,b,a+b$ always tile some interval. The subject has not received a great deal of further attention, although many interesting questions exist. All the information about tilings can be encoded in a finite digraph $D_{ab}$. We present several results about cycles and other structures in this graph. A number of conjectures and open problems are given.In [Go] an elegant proof by contradiction shows that a greedy algorithm will produce an interval tiling. We show that the process of converting to a direct proof leads to much stronger results.

scholarly journals On symmetric structures of order two

2008 ◽  
Vol Vol. 10 no. 2 (Combinatorics) ◽  
Author(s):  
Michel Bousquet ◽  
Cédric Lamathe
Keyword(s):  
Infinite Family ◽  
Combinatorial Objects ◽  
Integer Sequence ◽  
International Audience ◽  
Symmetric Structures

Combinatorics International audience Let (w_n)0 < n be the sequence known as Integer Sequence A047749 In this paper, we show that the integer w_n enumerates various kinds of symmetric structures of order two. We first consider ternary trees having a reflexive symmetry and we relate all symmetric combinatorial objects by means of bijection. We then generalize the symmetric structures and correspondences to an infinite family of symmetric objects.

scholarly journals Triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ and Tropical Oriented Matroids

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Suho Oh ◽  
Hwanchul Yoo
Keyword(s):  
Oriented Matroids ◽  
Oriented Matroid ◽  
Nous Proposons ◽  
Combinatorial Objects ◽  
New Class ◽  
Nous Montrons ◽  
International Audience ◽  
Tropical Polytopes

International audience Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes. Develin et Sturmfels ont montré que les triangulations de $\Delta_{n-1} \times \Delta_{d-1}$ peuvent être considérées comme des polytopes tropicaux. Les matroïdes orientés tropicaux ont été définis par Ardila et Develin, et ils ont été conjecturés être en bijection avec les subdivisions de $\Delta_{n-1} \times \Delta_{d-1}$. Dans cet article, nous montrons que toute triangulation de $\Delta_{n-1} \times \Delta_{d-1}$ encode un matroïde orienté tropical. De plus, nous proposons une nouvelle classe d'objets combinatoires qui peuvent décrire toutes les subdivisions d'une plus grande classe de polytopes.

scholarly journals A Formula for the Möbius Function of the Permutation Poset Based on a Topological Decomposition

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jason P Smith
Keyword(s):  
Significant Proportion ◽  
Möbius Function ◽  
Permutation Patterns ◽  
Mobius Function ◽  
International Audience ◽  
Definition Of ◽  
Object Of Study ◽  
Pattern Containment ◽  
Exponential Complexity ◽  
Exponential Part

International audience The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.

scholarly journals Constructing combinatorial operads from monoids

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Rooted Trees ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
International Audience ◽  
Motzkin Paths ◽  
Planar Rooted Trees

International audience We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

scholarly journals Classification of skew translation generalized quadrangles, I

2015 ◽  
Vol Vol. 17 no. 1 (Combinatorics) ◽  
Author(s):  
Koen Thas
Keyword(s):  
Generalized Quadrangles ◽  
New Classification ◽  
Open Problems ◽  
International Audience ◽  
Rank 2

Combinatorics International audience We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B2), more precisely in the (large) subtheory of skew translation generalized quadrangles (``STGQs''). Some of these involve, and solve, long-standing open problems.

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