scholarly journals Structure of Stable Sand Piles Model

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Thi Ha Duong Phan ◽  
Thi Thu Huong Tran
Keyword(s):  
Lattice Structure ◽  
Stable Configuration ◽  
Recursive Structure ◽  
Evolution Rule ◽  
Generating Tree ◽  
International Audience ◽  
Sand Piles ◽  
Infinite Set

International audience In this paper we study a variant of the Sand Piles Model, where the evolution rule consists of the falling down of one grain to a random column and an avalanche to reach a stable configuration. We prove that the infinite set of all stable configurations have a lattice structure which is a sublattice of Young lattice. At the end, based on a discussion about avalanches, we construct a generating tree of this model and show its strongtly recursive structure.

scholarly journals On a Class of Optimal Stopping Problems with Mixed Constraints

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
F. Thomas Bruss
Keyword(s):  
Selection Strategy ◽  
Optimal Selection ◽  
Recursive Structure ◽  
Limiting Behavior ◽  
Mixed Constraints ◽  
Selection Strategies ◽  
International Audience ◽  
Optimal Stopping Problems ◽  
The Cost ◽  
Independent Identically Distributed

International audience Let X(1),X(2),...,X(n) be independent, identically distributed uniform random variables on [0, 1]. We can observe the outcomes sequentially and must select online at least r of them, and, moreover, in expectation at least mu >= r. Here mu need not be integer. We see X(k) as the cost of selecting item k and want to minimize the expected total cost under the described combined (r, mu)-constraint. We will see that an optimal selection strategy exists on the set S(n) of all selection strategies for which the decision at instant k may depend on the value X(k), on the number N(k) of selections up to time k and of the number n - k of forthcoming observations. Let sigma(r,mu)(n) be the corresponding S(n)-optimal selection strategy and v(r,mu)(n) its value. The main goal of this paper is to determine these and to understand the limiting behavior of v(r,mu)(n). After discussion of the specific character of this combination of two types of constraints we conclude that the S(n)-problem has a recursive structure and solve it in terms of a double recursion. Our interest will then focus on the limiting behavior of nv(r,mu)(n) as n -> infinity. This sequence converges and its limit allows for the interpretation of a normalized limiting cost L (r, mu) of the (r, mu)-constraint. Our main result is that L(r, mu) = g(r) ((mu - r)(2)/(2)) where g(r) is the r(th) iterate of the function g(x) = 1 + x + root 1 + 2x. Our motivation to study mixed-constraints problems is indicated by several examples of possible applications. We also shortly discuss the intricacy of the expectational part of the constraint if we try to extend the class of strategies S n to the set of full-history-dependent and/or randomized strategies.

scholarly journals Partitions of an Integer into Powers

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Matthieu Latapy
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Tree Structure ◽  
Dynamical Model ◽  
International Audience ◽  
Partitions Of Integers ◽  
Natural Way

International audience In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

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 Generating trees for partitions and permutations with no k-nestings

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sophie Burrill ◽  
Sergi Elizalde ◽  
Marni Mishna ◽  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Young Tableaux ◽  
Set Partitions ◽  
Lattice Walks ◽  
Generating Trees ◽  
Generating Tree ◽  
International Audience ◽  
Tree Approach

International audience We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter. Nous décrivons une approche, basée sur l'utilisation d'arbres de génération, pour énumération et la génération exhaustive de partitions et permutations sans k-emboîtement. Contrairement aux travaux antérieurs qui reposent sur un lien entre ces objets, tableaux de Young et familles de chemins dans des treillis, notre approche traite directement partitions et diagrammes de permutations. Nous fournissons des équations fonctionnelles explicites pour les séries génératrices, avec k en tant que paramètre.

scholarly journals Linear recognition of generalized Fibonacci cubes $Q_h(111)$

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Yoomi Rho ◽  
Aleksander Vesel
Keyword(s):  
Linear Time ◽  
Binary String ◽  
Recursive Structure ◽  
Fibonacci Cube ◽  
Vertex Set ◽  
International Audience ◽  
Binary Strings

International audience The generalized Fibonacci cube $Q_h(f)$ is the graph obtained from the $h$-cube $Q_h$ by removing all vertices that contain a given binary string $f$ as a substring. In particular, the vertex set of the 3rd order generalized Fibonacci cube $Q_h(111)$ is the set of all binary strings $b_1b_2 \ldots b_h$ containing no three consecutive 1's. We present a new characterization of the 3rd order generalized Fibonacci cubes based on their recursive structure. The characterization is the basis for an algorithm which recognizes these graphs in linear time.

scholarly journals Lattice of combinatorial Hopf algebras: binary trees with multiplicities

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jean-Baptiste Priez
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Lattice Structure ◽  
Binary Trees ◽  
Hook Length ◽  
Combinatorial Hopf Algebras ◽  
Combinatorial Hopf Algebra ◽  
International Audience ◽  
Comme Application ◽  
Premiere Partie

International audience In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees. Dans une première partie, nous formalisons la construction d’algèbres de Hopf combinatoires à partir d’une réalisation polynomiale et de monoïdes de type monoïde plaxique. Grâce à cette construction, nous mettons à jour une structure de treillis sur ces algèbres de Hopf combinatoires. Comme application, nous construisons une nouvelle algèbre de Hopf sur des arbres binaires à multiplicités et on l’utilise pour démontrer une formule des équerressur ces arbres.

scholarly journals The representation of the symmetric group on $m$-Tamari intervals (conference version)

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Guillaume Chapuy ◽  
Louis-François Préville-Ratelle
Keyword(s):  
Symmetric Group ◽  
Lattice Structure ◽  
Parking Functions ◽  
Tamari Lattice ◽  
The North ◽  
Starting Point ◽  
International Audience ◽  
Associated Representation ◽  
Frobenius Series ◽  
Refined Version

International audience An $m$-ballot path of size $n$ is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at $(mn,n)$, and never going below the line $\{x=my\}$. The set of these paths can be equipped with a lattice structure, called the $m$-Tamari lattice and denoted by $\mathcal{T}{_n}^{(m)}$, which generalizes the usual Tamari lattice $\mathcal{T}n$ obtained when $m=1$. This lattice was introduced by F. Bergeron in connection with the study of diagonally coinvariant spaces in three sets of $n$ variables. The representation of the symmetric group $\mathfrak{S}_n$ on these spaces is conjectured to be closely related to the natural representation of $\mathfrak{S}_n$ on (labelled) intervals of the $m$-Tamari lattice studied in this paper. An interval $[P,Q$] of $\mathcal{T}{_n}^{(m)}$ is labelled if the north steps of $Q$ are labelled from 1 to $n$ in such a way the labels increase along any sequence of consecutive north steps. The symmetric group $\mathfrak{S}_n$ acts on labelled intervals of $\mathcal{T}{_n}^{(m)}$by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of $\mathfrak{S}_n$. In particular, the dimension of the representation, that is, the number of labelled $m$-Tamari intervals of size $n$, is found to be $(m+1)^n(mn+1)^{n-2}$. These results are new, even when $m=1$. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of $m$-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. The form of this equation is highly non-standard: it involves two additional variables $x$ and $y$, a derivative with respect to $y$ and iterated divided differences with respect to $x$. The hardest part of the proof consists in solving it, and we develop original techniques to do so.

scholarly journals Slicings of parallelogram polyominoes, or how Baxter and Schröder can be reconciled

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Lattice Paths ◽  
Schröder Numbers ◽  
Generating Tree ◽  
International Audience

International audience We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes (called slicings) which grow according to these succession rules. We also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, and a new Schröder subset of Baxter permutations.

scholarly journals Projections et cohérence de vues dans les grammaires algébriques

2008 ◽  
Vol Volume 8, Special Issue... ◽  
Author(s):  
Eric Badouel ◽  
Maurice Tchoupé Tchendji
Keyword(s):  
Global Structure ◽  
Lazy Evaluation ◽  
Document Structure ◽  
Legal Structures ◽  
International Audience ◽  
Structured Document ◽  
Partial View ◽  
Context Free ◽  
Infinite Set ◽  
Structural Aspects

International audience A complex structured document is intentionnally represented as a tree decorated with attributes. The set of legal structures is given by an abstract context-free grammar. We forget about the attributes; they are related with semantical issues that can be treated independently of the purely structural aspects that we address in this article. That intentional representation may be asynchronously manipulated by a set of independent tools each of which operates on a distinct partial view of the whole structure. In order to synchronize these various partial views, we are faced to the problem of their coherence: can we decide whether there exists some global structure corresponding to a given set of partial views and in the affirmative, can we produce such a global structure ? We solve this problem in the case where a view is given by a subset of grammatical symbols, those associated with the so-called visible syntactical categories. The proposed algorithm, that strongly relies on the mechanism of lazy evaluation, produces an answer to this problem even if partial views may correspond to an infinite set of related global structures. Un document structuré complexe est représenté intentionnellement sous la forme d'une structure arborescente décorée par des attributs. Les structures licites sont caractérisées par une grammaire algébrique abstraite. Nous faisons ici abstraction des attributs ; ces derniers sont liés à des aspects sémantiques qui peuvent être traités séparément des aspects purement structurels qui nous intéressent ici. Cette représentation intentionnelle peut être manipulée de façon indépendante et éventuellement non synchronisée par divers outils d'édition et de manipulation qui opèrent sur des vues partielles distinctes du même document. Pour la re-synchronisation de ces vues partielles nous devons résoudre le problème de leur cohérence : décider s'il existe un document correspondant à ces différentes vues et dans l'affirmative produire un tel document. Nous montrons comment résoudre ce problème dans le cas où chaque vue est associée à un sous-ensemble des symboles grammaticaux : ceux qui correspondent aux catégories syntaxiques visibles. L'algorithme proposé, qui repose fortement sur le mécanisme d'évaluation paresseuse, résout ce problème même dans le cas où chaque vue partielle correspond à un nombre infini de documents possibles.

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