scholarly journals Two Pile Move-Size Dynamic Nim

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Arthur Holshouser ◽  
Harold Reiter
Keyword(s):  
Special Class ◽  
Winning Strategy ◽  
Maximum Size ◽  
Single Pile ◽  
International Audience

International audience The purpose of this paper is to solve a special class of combinational games consisting of two-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the games. Two players alternate moving. Each player in his turn first chooses one of the piles, and his choice of piles can change from move to move. He then removes counters from this chosen pile. A function f:Z^+ → Z^+ is given which determines the maximum size of the next move in terms of the current move size. The game ends as soon as one of the two piles is empty, and the winner is the last player to move in the game. The games for which f(k)=k, f(k)=2k, and f(k)=3k use the same formula for computing the smallest winning move size. Here we find all the functions f for which this formula works, and we also give the winning strategy for each function. See Holshouser, A, James Rudzinski and Harold Reiter: Dynamic One-Pile Nim for a discussion of the single pile game.

scholarly journals The game of arboricity

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Tomasz Bartnicki ◽  
Jaroslaw Grytczuk ◽  
Hal Kierstead
Keyword(s):  
Upper Bound ◽  
Winning Strategy ◽  
Minimum Size ◽  
Vertex Coloring ◽  
Fixed Set ◽  
International Audience

International audience Using a fixed set of colors $C$, Ann and Ben color the edges of a graph $G$ so that no monochromatic cycle may appear. Ann wins if all edges of $G$ have been colored, while Ben wins if completing a coloring is not possible. The minimum size of $C$ for which Ann has a winning strategy is called the $\textit{game arboricity}$ of $G$, denoted by $A_g(G)$. We prove that $A_g(G) \leq 3k$ for any graph $G$ of arboricity $k$, and that there are graphs such that $A_g(G) \geq 2k-2$. The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide other strategie based on induction.

scholarly journals A randomized algorithm for finding a maximum clique in the visibility graph of a simple polygon

2015 ◽  
Vol Vol. 17 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Sergio Cabello ◽  
Maria Saumell
Keyword(s):  
Hamiltonian Cycle ◽  
Randomized Algorithm ◽  
Maximum Clique ◽  
Simple Polygon ◽  
Maximum Size ◽  
Visibility Graph ◽  
Probability Of Error ◽  
Discrete Algorithms ◽  
International Audience ◽  
Previous Algorithm

Discrete Algorithms International audience We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle describing the boundary of P, and a parameter δ∈(0,1) controlling the probability of error of the algorithm. The algorithm does not require the coordinates of the vertices of P. With probability at least 1-δ the algorithm runs in O( |E(G)|2 / ω(G) log(1/δ)) time and returns a maximum clique, where ω(G) is the number of vertices in a maximum clique in G. A deterministic variant of the algorithm takes O(|E(G)|2) time and always outputs a maximum size clique. This compares well to the best previous algorithm by Ghosh et al. (2007) for the problem, which is deterministic and runs in O(|V(G)|2 |E(G)|) time.

scholarly journals Combinatorics of Positroids

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Suho Oh
Keyword(s):  
Special Class ◽  
Combinatorial Description ◽  
Combinatorial Objects ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets

International audience Recently Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. There are many interesting combinatorial objects associated to a positroid. We introduce some recent results, including the generalization and proof of the purity conjecture by Leclerc and Zelevinsky on weakly separated sets.

scholarly journals Properties of Random Graphs via Boltzmann Samplers

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Konstantinos Panagiotou ◽  
Andreas Weißl
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Maximum Size ◽  
Structural Constraints ◽  
Convex Polygons ◽  
Typical Member ◽  
Graph Classes ◽  
International Audience ◽  
Cactus Graphs

International audience This work is devoted to the understanding of properties of random graphs from graph classes with structural constraints. We propose a method that is based on the analysis of the behaviour of Boltzmann sampler algorithms, and may be used to obtain precise estimates for the maximum degree and maximum size of a biconnected block of a "typical'' member of the class in question. We illustrate how our method works on several graph classes, namely dissections and triangulations of convex polygons, embedded trees, and block and cactus graphs.

scholarly journals On additive combinatorics of permutations of ℤn

2014 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
L. Sunil Chandran ◽  
Deepak Rajendraprasad ◽  
Nitin Singh
Keyword(s):  
Maximum Size ◽  
Extremal Problems ◽  
Additive Combinatorics ◽  
Special Issue ◽  
Prime Divisors ◽  
Ring Of Integers ◽  
Euler’S Totient Function ◽  
International Audience ◽  
Addition And Subtraction

Special issue PRIMA 2013 International audience Let ℤ<sub>n</sub> denote the ring of integers modulo n. A permutation of ℤ<sub>n</sub> is a sequence of n distinct elements of ℤ<sub>n</sub>. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤ<sub>n</sub>, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s(n) and t(n) respectively. The case when n is even is trivial in both the cases, with s(n)=1 and t(n)=n!. For n odd, we prove (nφ(n))/2<sup>k</sup>≤s(n)≤n!· 2<sup>-(n-1)/2</sup>((n-1)/2)! and 2<sup>(n-1)/2</sup>·(n-1 / 2)!≤t(n)≤ 2<sup>k</sup>·(n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function.

scholarly journals Bijections on m-level Rook Placements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Kenneth Barrese ◽  
Bruce Sagan
Keyword(s):  
Special Class ◽  
Symmetric Functions ◽  
Elementary Symmetric Functions ◽  
Rook Placements ◽  
Rook Numbers ◽  
Ferrers Board ◽  
International Audience ◽  
Ferrers Boards

International audience Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of $m$-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne. Nous considérons les rangs d’un échiquier partagés en ensembles de $m$ rangs appelés les niveaux. Un $m$-placement des tours est un sous-ensemble des carrés du plateau tel qu’il n’y a pas deux carrés dans la même colonne ou dans le même niveau. Nous construisons deux bijections explicites entre des plateaux de Ferrers ayant les mêmes nombres de $m$-placements. La première est une généralisation d’une fonction de Foata et Schützenberger et notre démonstration est pour n’importe quels plateaux de Ferrers. La deuxième généralise une bijection de Loehr et Remmel. Cette construction marche seulement pour des plateaux particuliers, mais ça donne une formule pour le nombre de $m$-placements en terme des fonctions symétriques élémentaires. Enfin, nous généralisons un autre résultat de Loehr et Remmel donnant une bijection entre deux plateaux ayant les mêmes nombres de coups. Les deux dernières bijections utilisent le Principe des Involutions de Garsia et Milne.

scholarly journals Tropical Ideals

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Diane Maclagan ◽  
Felipe Rincón
Keyword(s):  
Special Class ◽  
Tropical Geometry ◽  
Combinatorial Structure ◽  
International Audience ◽  
Algebraic Foundation

International audience We introduce and study a special class of ideals over the semiring of tropical polynomials, which we calltropical ideals, with the goal of developing a useful and solid algebraic foundation for tropical geometry. We exploretheir rich combinatorial structure, and prove that they satisfy numerous properties analogous to classical ideals.

scholarly journals Polyominoes determined by permutations

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
I. Fanti ◽  
A. Frosini ◽  
E. Grazzini ◽  
R. Pinzani ◽  
S. Rinaldi
Keyword(s):  
Special Class ◽  
International Audience

International audience In this paper we consider the class of $\textit{permutominoes}$, i.e. a special class of polyominoes which are determined by a pair of permutations having the same size. We give a characterization of the permutations associated with convex permutominoes, and then we enumerate various classes of convex permutominoes, including parallelogram, directed-convex, and stack ones.

scholarly journals On an open problem of Green and Losonczy: exact enumeration of freely braided permutations

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Toufik Mansour
Keyword(s):  
Representation Theory ◽  
Generating Function ◽  
Coxeter Group ◽  
Upper Bound ◽  
Open Problem ◽  
Special Class ◽  
Exact Enumeration ◽  
Restricted Permutations ◽  
International Audience ◽  
Freely Braided

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

scholarly journals Pairwise Intersections and Forbidden Configurations

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Richard P. Anstee ◽  
Peter Keevash
Keyword(s):  
Complete Intersection ◽  
Maximum Size ◽  
Strong Stability ◽  
Independent Interest ◽  
Intersection Theorem ◽  
International Audience ◽  
Forbidden Configurations ◽  
Element Set

International audience Let $f_m(a,b,c,d)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $m$-element set for which there is no pair of subsets $A,B \in \mathcal{F}$ with $|A \cap B| \geq a$, $|\bar{A} \cap B| \geq b$, $|A \cap \bar{B}| \geq c$, and $|\bar{A} \cap \bar{B}| \geq d$. By symmetry we can assume $a \geq d$ and $b \geq c$. We show that $f_m(a,b,c,d)$ is $\Theta (m^{a+b-1})$ if either $b > c$ or $a,b \geq 1$. We also show that $f_m(0,b,b,0)$ is $\Theta (m^b)$ and $f_m(a,0,0,d)$ is $\Theta (m^a)$. This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.

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