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 Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

10.37236/2001 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Petter Brändén ◽  
Anders Claesson
Keyword(s):  
Fixed Points ◽  
Relative Position ◽  
Permutation Statistics ◽  
Permutation Patterns ◽  
Alternating Permutations ◽  
Major Index ◽  
Permutation Pattern ◽  
New Type ◽  
Permutation Statistic ◽  
Infinite Linear

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

scholarly journals Tiling Periodicity

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Juhani Karhumaki ◽  
Yury Lifshits ◽  
Wojciech Rytter
Keyword(s):  
Open Problem ◽  
Classical Case ◽  
Side Product ◽  
Compact Representation ◽  
Minimal Size ◽  
Partial Word ◽  
International Audience ◽  
New Type

International audience We contribute to combinatorics and algorithmics of words by introducing new types of periodicities in words. A tiling period of a word w is partial word u such that w can be decomposed into several disjoint parallel copies of u, e.g. a lozenge b is a tiling period of a a b b. We investigate properties of tiling periodicities and design an algorithm working in O(n log (n) log log (n)) time which finds a tiling period of minimal size, the number of such minimal periods and their compact representation. The combinatorics of tiling periods differs significantly from that for classical full periods, for example unlike the classical case the same word can have many different primitive tiling periods. We consider also a related new type of periods called in the paper multi-periods. As a side product of the paper we solve an open problem posted by T. Harju (2003).

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 Chromatic Turán problems and a new upper bound for the Turán density of $\mathcal{K}_4^-$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
John Talbot
Keyword(s):  
Upper Bound ◽  
Turan Problems ◽  
International Audience ◽  
New Type ◽  
Extremal Hypergraph

International audience We consider a new type of extremal hypergraph problem: given an $r$-graph $\mathcal{F}$ and an integer $k≥2$ determine the maximum number of edges in an $\mathcal{F}$-free, $k$-colourable $r$-graph on $n$ vertices. Our motivation for studying such problems is that it allows us to give a new upper bound for an old problem due to Turán. We show that a 3-graph in which any four vertices span at most two edges has density less than $\frac{33}{ 100}$, improving previous bounds of $\frac{1}{ 3}$ due to de Caen [1], and $\frac{1}{ 3}-4.5305×10^-6$ due to Mubayi [9].

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 Biased weak polyform achievement games

2014 ◽  
Vol Vol. 16 no. 3 (Combinatorics) ◽  
Author(s):  
Ian Norris ◽  
Nándor Sieben
Keyword(s):  
Winning Strategies ◽  
International Audience ◽  
New Type

Combinatorics International audience In a biased weak (a,b) polyform achievement game, the maker and the breaker alternately mark a,b previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a,b) game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all (a,b) pairs for polyiamonds and polyominoes up to size four.

scholarly journals Place-Difference-Value Patterns: A Generalization of Generalized Permutation and Word Patterns

Integers ◽  
2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel
Keyword(s):  
Partially Ordered Sets ◽  
General Definition ◽  
Permutation Patterns ◽  
Ordered Sets ◽  
Combinatorial Objects ◽  
Permutation Pattern ◽  
New Type ◽  
Number Of Publications ◽  
Definition Of ◽  
Patterns In Permutations

AbstractMotivated by the study of Mahonian statistics, in 2000, Babson and Steingrímsson [Sém. Lothar. Comb] introduced the notion of a “generalized permutation pattern” (GP) which generalizes the concept of “classical” permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingrímsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-Mélou et al. [J. Comb. Theory A] introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets.In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs provide many new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we discuss several bijective questions linking our patterns to other combinatorial objects.

scholarly journals Scheduling Problems and Generalized Graph Coloring

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
John Machacek
Keyword(s):  
Graph Coloring ◽  
Symmetric Function ◽  
Vertex Coloring ◽  
Scheduling Problems ◽  
International Audience ◽  
New Type ◽  
Chromatic Symmetric Function ◽  
Special Case

International audience We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, andsimplicial complexes. To this coloring there is an associated symmetric function in noncommuting variables for whichwe give a deletion-contraction formula. In the case of graphs our symmetric function in noncommuting variablesagrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloringis a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contractionlaw can be applied to scheduling problems.

scholarly journals Weak Positional Games on Hypergraphs of Rank Three

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Martin Kutz
Keyword(s):  
Polynomial Time ◽  
Optimal Strategies ◽  
Decomposition Theorem ◽  
Positional Games ◽  
Almost Disjoint ◽  
International Audience ◽  
New Type

International audience In a weak positional game, two players, Maker and Breaker, alternately claim vertices of a hypergraph until either Maker wins by getting a complete edge or all vertices are taken without this happening, a Breaker win. For the class of almost-disjoint hypergraphs of rank three (edges with up to three vertices only and edge-intersections on at most one vertex) we show how to find optimal strategies in polynomial time. Our result is based on a new type of decomposition theorem which might lead to a better understanding of weak positional games in general.

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