scholarly journals Combinatorial aspects of pyramids of one-dimensional pieces of fixed integer length

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Bergfinnur Durhuus ◽  
Søren Eilers
Keyword(s):  
Combinatorial Proof ◽  
Average Width ◽  
Fixed Integer ◽  
Bijective Correspondence ◽  
One Dimensional ◽  
International Audience

International audience We consider pyramids made of one-dimensional pieces of fixed integer length $a$ and which may have pairwise overlaps of integer length from $1$ to $a$. We give a combinatorial proof that the number of pyramids of size $m$, i.e., consisting of $m$ pieces, equals $\binom{am-1}{m-1}$ for each $a \geq 2$. This generalises a well known result for $a=2$. A bijective correspondence between so-called right (or left) pyramids and $a$-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids equals $\sqrt{\frac{\pi}{2} a(a-1)m}$.

scholarly journals Monodromy and K-theory of Schubert curves via generalized jeu de taquin

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maria Monks Gillespie ◽  
Jake Levinson
Keyword(s):  
Local Algorithm ◽  
Jeu De Taquin ◽  
Bijective Correspondence ◽  
One Dimensional ◽  
The Real ◽  
Real Locus ◽  
Real Geometry ◽  
International Audience ◽  
K Theory ◽  
Rational Normal Curve

International audience We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of RP1, with ω as the monodromy operator.We provide a fast, local algorithm for computing ω without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.

scholarly journals Random sampling of lattice paths with constraints, via transportation

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Lucas Gerin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Sampling ◽  
Optimal Transport ◽  
Lattice Paths ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Contraction Property ◽  
International Audience

International audience We build and analyze in this paper Markov chains for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. These chains are easy to implement, and sample an "almost" uniform path of length $n$ in $n^{3+\epsilon}$ steps. This bound makes use of a certain $\textit{contraction property}$ of the Markov chain, and is proved with an approach inspired by optimal transport.

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 Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Stirling Numbers ◽  
Algebraic Methods ◽  
Combinatorial Proof ◽  
Nous Obtenons ◽  
Long Cycle ◽  
Bijective Proof ◽  
International Audience

International audience We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account. Nous étudions le nombre de permutations $\beta$ de $[N]$ avec $m$ cycles telles que $(1 2 \ldots N) \beta^{-1}$ a un seul cycle. Ces nombres apparaissent en tant que coefficients des monômes linéaires des polynômes de Kerov et de Stanley. À l'aide de méthodes algébriques, D. Zagier a trouvé une connexion inattendue avec les nombres de Stirling de taille $N+1$. Nous présentons ici la première preuve combinatoire de son résultat, en introduisant une nouvelle bijection entre des cartes partitionnées et des arbres épineux. De plus, nous obtenons un résultat plus fin, prenant en compte le type des permutations.

scholarly journals A stronger recognizability condition for two-dimensional languages

2013 ◽  
Vol Vol. 15 no. 2 (Automata, Logic and Semantics) ◽  
Author(s):  
Marcella Anselmo ◽  
Maria Madonia
Keyword(s):  
Finite Automata ◽  
The Other ◽  
Two Dimensional ◽  
Minimal Size ◽  
One Dimensional ◽  
Tiling System ◽  
International Audience ◽  
The One

Automata, Logic and Semantics International audience The paper presents a condition necessarily satisfied by (tiling system) recognizable two-dimensional languages. The new recognizability condition is compared with all the other ones known in the literature (namely three conditions), once they are put in a uniform setting: they are stated as bounds on the growth of some complexity functions defined for two-dimensional languages. The gaps between such functions are analyzed and examples are shown that asymptotically separate them. Finally the new recognizability condition results to be the strongest one, while the remaining ones are its particular cases. The problem of deciding whether a two-dimensional language is recognizable is here related to the one of estimating the minimal size of finite automata recognizing a sequence of (one-dimensional) string languages.

scholarly journals Projective subdynamics and universal shifts

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Pierre Guillon
Keyword(s):  
Large Class ◽  
Finite Type ◽  
Positive Entropy ◽  
Two Dimensional ◽  
One Dimensional ◽  
International Audience

International audience We study the projective subdynamics of two-dimensional shifts of finite type, which is the set of one-dimensional configurations that appear as columns in them. We prove that a large class of one-dimensional shifts can be obtained as such, namely the effective subshifts which contain positive-entropy sofic subshifts. The proof involves some simple notions of simulation that may be of interest for other constructions. As an example, it allows us to prove the undecidability of all non-trivial properties of projective subdynamics.

scholarly journals The analysis of a prioritised probabilistic algorithm to find large induced forests in regular graphs with large girth

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Carlos Hoppen
Keyword(s):  
Lower Bounds ◽  
Regular Graphs ◽  
Probabilistic Algorithms ◽  
Induced Subgraph ◽  
Fixed Integer ◽  
Asymptotic Bounds ◽  
Random Regular Graph ◽  
International Audience ◽  
Large Girth ◽  
Deterministic Bounds

International audience The analysis of probabilistic algorithms has proved to be very successful for finding asymptotic bounds on parameters of random regular graphs. In this paper, we show that similar ideas may be used to obtain deterministic bounds for one such parameter in the case of regular graphs with large girth. More precisely, we address the problem of finding a large induced forest in a graph $G$, by which we mean an acyclic induced subgraph of $G$ with a lot of vertices. For a fixed integer $r \geq 3$, we obtain new lower bounds on the size of a maximum induced forest in graphs with maximum degree $r$ and large girth. These bounds are derived from the solution of a system of differential equations that arises naturally in the analysis of an iterative probabilistic procedure to generate an induced forest in a graph. Numerical approximations suggest that these bounds improve substantially the best previous bounds. Moreover, they improve previous asymptotic lower bounds on the size of a maximum induced forest in a random regular graph.

scholarly journals Atomic Bases and $T$-path Formula for Cluster Algebras of Type $D$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Emily Gunawan ◽  
Gregg Musiker
Keyword(s):  
Cluster Algebra ◽  
Cluster Algebras ◽  
Combinatorial Proof ◽  
Expansion Formula ◽  
Type D ◽  
International Audience ◽  
T Path ◽  
Nous Utilisons

International audience We extend a $T$-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type $D$. Nous généralisons une formule de développement en $T$-chemins pour les arcs sur une surface non-perforée aux arcs sur un polygone à une perforation. Nous utilisons cette formule pour donner une preuve combinatoire du fait que les monômes amassées constituent la base atomique d’une algèbre amassée de type $D$.

scholarly journals A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Miguel Méndez ◽  
Adolfo Rodríguez
Keyword(s):  
Stirling Numbers ◽  
Direct Application ◽  
Bell Numbers ◽  
Bijective Correspondence ◽  
Rook Placements ◽  
Combinatorial Model ◽  
International Audience ◽  
Generalized Stirling Numbers ◽  
Dual Graded Graphs ◽  
Combinatorial Methods

International audience We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.

scholarly journals Counting self-dual interval orders

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vít Jelínek
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Interval Orders ◽  
Maximal Elements ◽  
Triangular Matrices ◽  
Bijective Correspondence ◽  
Upper Triangular Matrices ◽  
Positive Entry ◽  
International Audience ◽  
Natural Statistics

International audience In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry. Dans cet article, on obtient une expression explicite pour la fonction génératrice du nombre des ensembles partiellement ordonnés (posets) qui évitent le motif (2+2). La fonction compte ces ensembles par rapport à plusieurs statistiques naturelles, incluant le nombre d'éléments, le nombre de niveaux, et le nombre d'éléments minimaux et maximaux. Dans la deuxième partie, on obtient une expression similaire pour la fonction génératrice des posets autoduaux évitant le motif (2+2). On obtient ces résultats à l'aide d'une bijection entre les posets évitant (2+2) et les matrices triangulaires supérieures dont chaque ligne et chaque colonne contient un élément positif.

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