scholarly journals The absence of a pattern and the occurrences of another

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Miklós Bóna
Keyword(s):  
Exact Enumeration ◽  
Expected Number ◽  
Permutation Pattern ◽  
International Audience

International audience Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern q in permutations that avoid another given pattern r. In some cases, we find the pattern that occurs least often, (resp. most often) in all r-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising.

scholarly journals Stochastic Flips on Dimer Tilings

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Thomas Fernique ◽  
Damien Regnault
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Upper Bound ◽  
Numerical Experiments ◽  
Triangular Grid ◽  
Expected Number ◽  
Worst Case ◽  
Average Case ◽  
International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

scholarly journals Paths of specified length in random k-partite graphs

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
C.R. Subramanian
Keyword(s):  
Random Graphs ◽  
Expected Number ◽  
Vertex Set ◽  
International Audience ◽  
Simple Paths ◽  
Positive Integers

International audience Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.

scholarly journals Upper bounds on the non-3-colourability threshold of random graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Upper Bounds ◽  
Average Degree ◽  
Expected Number ◽  
International Audience ◽  
Full Analysis ◽  
A Minor ◽  
Minor Improvement

International audience We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.

scholarly journals Hopf algebra of permutation pattern functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Yannic Vargas
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Point Of View ◽  
Algebraic Combinatorics ◽  
Permutation Patterns ◽  
Permutation Pattern ◽  
International Audience ◽  
Pattern Functions

International audience We study permutation patterns from an algebraic combinatorics point of view. Using analogues of the classical shuffle and infiltration products for word, we define two new Hopf algebras of permutations related to the notion of permutation pattern. We show several remarkable properties of permutation patterns functions, as well their occurrence in other domains.

scholarly journals The topology of the permutation pattern poset

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Peter McNamara ◽  
Einar Steingrımsson
Keyword(s):  
Homotopy Type ◽  
Recursive Formula ◽  
Large Classes ◽  
Permutation Pattern ◽  
International Audience ◽  
Rooted Forest ◽  
Carry Over ◽  
Almost All ◽  
Separable Permutations ◽  
Pattern Containment

International audience The set of all permutations, ordered by pattern containment, forms a poset. This extended abstract presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.

scholarly journals Sorting and preimages of pattern classes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Anders Claesson ◽  
Henning Úlfarsson
Keyword(s):  
Permutation Pattern ◽  
International Audience ◽  
New Type ◽  
Standing Problem

International audience We introduce an algorithm to determine when a sorting operation, such as stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a new proof of the description of West-2-stack-sortable permutations, that is permutations that are completely sorted when passed twice through a stack, in terms of patterns. We also solve the long-standing problem of describing West-3-stack-sortable permutations. This requires a new type of generalized permutation pattern we call a decorated pattern. On introduit un algorithme qui détermine quand un opérateur de tri (tels que le tri par une pile, ou le tri-bulle) produit un motif donnè en sortie. Cet algorithme fournit une nouvelle preuve de la caractérisation des permutations 2-triables au sens de West (c'est-à-dire des permutations qui sont triées complètement par deux passages dans une pile) par des motifs interdits. On résout aussi le problème longtemps ouvert de la caractérisation des permutations 3-triables au sens de West. Ceci demande de définir un nouveau type de motifs généralisés, appelè motif décoré.

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 Periodic Patterns of Signed Shifts

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Kassie Archer ◽  
Sergi Elizalde
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Relative Order ◽  
Exact Enumeration ◽  
Periodic Patterns ◽  
Tent Map ◽  
One Dimensional ◽  
Important Family ◽  
International Audience ◽  
Descent Set

International audience The periodic patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial description of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain transformation of the pattern. Signed shifts are an important family of one-dimensional dynamical systems. For particular types of signed shifts, namely shift maps, reverse shift maps, and the tent map, we give exact enumeration formulas for their periodic patterns. As a byproduct of our work, we recover some results of Gessel and Reutenauer and obtain new results on the enumeration of pattern-avoiding cycles.

scholarly journals A Probabilistic Counting Lemma for Complete Graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Stefanie Gerke ◽  
Martin Marciniszyn ◽  
Angelika Steger
Keyword(s):  
Random Graphs ◽  
Regular Graph ◽  
Complete Graphs ◽  
Regularity Lemma ◽  
Expected Number ◽  
International Audience ◽  
Partition Class ◽  
Szemeredi's Regularity Lemma ◽  
Almost All ◽  
Counting Lemma

International audience We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi's Regularity Lemma. More precisely, we consider the number of complete graphs $K_{\ell}$ on $\ell$ vertices in $\ell$-partite graphs where each partition class consists of $n$ vertices and there is an $\varepsilon$-regular graph on $m$ edges between any two partition classes. We show that for all $\beta > $0, at most a $\beta^m$-fraction of graphs in this family contain less than the expected number of copies of $K_{\ell}$ provided $\varepsilon$ is sufficiently small and $m \geq Cn^{2-1/(\ell-1)}$ for a constant $C > 0$ and $n$ sufficiently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Łuczak and Rödl and has several implications for random graphs.

scholarly journals Expected number of locally maximal solutions for random Boolean CSPs

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Nadia Creignou ◽  
Hervé Daudé ◽  
Olivier Dubois
Keyword(s):  
Large Class ◽  
Constraint Satisfaction ◽  
Upper Bound ◽  
Constraint Satisfaction Problems ◽  
Expected Number ◽  
Exponential Order ◽  
Maximal Solutions ◽  
International Audience ◽  
Locally Maximal ◽  
General Tool

International audience For a large number of random Boolean constraint satisfaction problems, such as random $k$-SAT, we study how the number of locally maximal solutions evolves when constraints are added. We give the exponential order of the expected number of these distinguished solutions and prove it depends on the sensitivity of the allowed constraint functions only. As a by-product we provide a general tool for computing an upper bound of the satisfiability threshold for any problem of a large class of random Boolean CSPs.

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