scholarly journals Avoiding patterns in irreducible permutations

2016 ◽  
Vol Vol. 17 no. 3 (Combinatorics) ◽  
Author(s):  
Jean-Luc Baril
Keyword(s):  
Fixed Point ◽  
Pattern Avoidance ◽  
Dyck Paths ◽  
Classical Pattern ◽  
International Audience ◽  
Motzkin Paths

International audience We explore the classical pattern avoidance question in the case of irreducible permutations, <i>i.e.</i>, those in which there is no index $i$ such that $\sigma (i+1) - \sigma (i)=1$. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length $n-1$ and the sets of irreducible permutations of length $n$ (respectively fixed point free irreducible involutions of length $2n$) avoiding a pattern $\alpha$ for $\alpha \in \{132,213,321\}$. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.

scholarly journals Constructing combinatorial operads from monoids

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Rooted Trees ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
International Audience ◽  
Motzkin Paths ◽  
Planar Rooted Trees

International audience We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

scholarly journals Minimal and maximal plateau lengths in Motzkin paths

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Helmut Prodinger ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Gumbel Distribution ◽  
Minimal Length ◽  
Analytic Method ◽  
Horizontal Segment ◽  
Motzkin Path ◽  
Dyck Paths ◽  
Substitution Process ◽  
International Audience ◽  
Motzkin Paths

International audience The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters <i> maximal plateau length, horizontal segment </i>and <i>maximal horizontal segment </i>as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.

scholarly journals Pattern-avoiding Dyck paths

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Antonio Bernini ◽  
Luca Ferrari ◽  
Renzo Pinzani ◽  
Julian West
Keyword(s):  
Asymptotic Behavior ◽  
Pattern Avoidance ◽  
Lattice Paths ◽  
Dyck Path ◽  
Typical Pattern ◽  
Open Problems ◽  
Dyck Paths ◽  
International Audience ◽  
Pattern Containment

International audience We introduce the notion of $\textit{pattern}$ in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the $\textit{Dyck pattern poset}$. Given a Dyck path $P$, we determine a formula for the number of Dyck paths covered by $P$, as well as for the number of Dyck paths covering $P$. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. Finally, we offer a conjecture concerning the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern and we pose a series of open problems regarding the structure of the Dyck pattern poset.

scholarly journals Pattern Avoidance in Task-Precedence Posets

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Mitchell Paukner ◽  
Lucy Pepin ◽  
Manda Riehl ◽  
Jarred Wieser
Keyword(s):  
Generating Function ◽  
Pattern Avoidance ◽  
Interesting Application ◽  
Hasse Diagrams ◽  
Dyck Paths ◽  
Classical Pattern ◽  
Multiple Task

We have extended classical pattern avoidance to a new structure: multiple task-precedence posets whose Hasse diagrams have three levels, which we will call diamonds. The vertices of each diamond are assigned labels which are compatible with the poset. A corresponding permutation is formed by reading these labels by increasing levels, and then from left to right. We used Sage to form enumerative conjectures for the associated permutations avoiding collections of patterns of length three, which we then proved. We have discovered a bijection between diamonds avoiding 132 and certain generalized Dyck paths. We have also found the generating function for descents, and therefore the number of avoiders, in these permutations for the majority of collections of patterns of length three. An interesting application of this work (and the motivating example) can be found when task-precedence posets represent warehouse package fulfillment by robots, in which case avoidance of both 231 and 321 ensures we never stack two heavier packages on top of a lighter package. Comment: 17 pages

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 Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

scholarly journals Grasshopper Avoidance of Patterns

10.37236/6210 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Michał Dębski ◽  
Urszula Pastwa ◽  
Krzysztof Węsek
Keyword(s):  
Euclidean Plane ◽  
Pattern Avoidance ◽  
Type Problem ◽  
Compression Method ◽  
New Variant ◽  
Classical Pattern ◽  
Entropy Compression ◽  
Line P ◽  
Entropy Compression Method

Motivated by a geometrical Thue-type problem, we introduce a new variant of the classical pattern avoidance in words, where jumping over a letter in the pattern occurrence is allowed. We say that pattern $p\in E^+$ occurs with jumps in a word $w=a_1a_2\ldots a_k \in A^+$, if there exist a non-erasing morphism $f$ from $E^*$ to $A^*$ and a sequence $(i_1, i_2, \ldots , i_l)$ satisfying $i_{j+1}\in\{ i_j+1, i_j+2 \}$ for $j=1, 2, \ldots, l-1$, such that $f(p) = a_{i_1}a_{i_2}\ldots a_{i_l}.$ For example, a pattern $xx$ occurs with jumps in a word $abdcadbc$ (for $x \mapsto abc$). A pattern $p$ is grasshopper $k$-avoidable if there exists an alphabet $A$ of $k$ elements, such that there exist arbitrarily long words over $A$ in which $p$ does not occur with jumps. The minimal such $k$ is the grasshopper avoidability index of $p$. It appears that this notion is related to two other problems: pattern avoidance on graphs and pattern-free colorings of the Euclidean plane. In particular, we show that a sequence avoiding a pattern $p$ with jumps can be a tool to construct a line $p$-free coloring of $\mathbb{R}^2$.    In our work, we determine the grasshopper avoidability index of patterns $\alpha^n$ for all $n$ except $n=5$. We also show that every doubled pattern is grasshopper $(2^7+1)$-avoidable, every pattern on $k$ variables of length at least $2^k$ is grasshopper $37$-avoidable, and there exists a constant $c$ such that every pattern of length at least $c$ on $2$ variables is grasshopper $3$-avoidable (those results are proved using the entropy compression method).

scholarly journals Pattern avoidance in alternating permutations and tableaux (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Alternating Permutations ◽  
Special Cases ◽  
Nous Montrons ◽  
Insertion Algorithm ◽  
Generating Trees ◽  
International Audience ◽  
Standard Young Tableaux

International audience We give bijective proofs of pattern-avoidance results for a class of permutations generalizing alternating permutations. The bijections employed include a modified form of the RSK insertion algorithm and recursive bijections based on generating trees. As special cases, we show that the sets $A_{2n}(1234)$ and $A_{2n}(2143)$ are in bijection with standard Young tableaux of shape $\langle 3^n \rangle$. Alternating permutations may be viewed as the reading words of standard Young tableaux of a certain skew shape. In the last section of the paper, we study pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape $\lambda / \mu$ whose reading words avoid $213$ is a natural $\mu$-analogue of the Catalan numbers. Similar results for the patterns $132$, $231$ and $312$. Nous présentons des preuves bijectives de résultats pour une classe de permutations à motifs exclus qui généralisent les permutations alternantes. Les bijections utilisées reposent sur une modification de l'algorithme d'insertion "RSK" et des bijections récursives basées sur des arbres de génération. Comme cas particuliers, nous montrons que les ensembles $A_{2n}(1234)$ et $A_{2n}(2143)$ sont en bijection avec les tableaux standards de Young de la forme $\langle 3^n \rangle$. Une permutation alternante peut être considérée comme le mot de lecture de certain skew tableau. Dans la dernière section de l'article, nous étudions l'évitement des motifs dans les mots de lecture de skew tableaux généraux. Nous montrons bijectivement que le nombre de tableaux standards de forme $\lambda / \mu$ dont les mots de lecture évitent $213$ est un $\mu$-analogue naturel des nombres de Catalan. Des résultats analogues sont valables pour les motifs $132$, $231$ et $312$.

scholarly journals Indecomposable permutations with a given number of cycles

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Robert Cori ◽  
Claire Mathieu
Keyword(s):  
Fixed Point ◽  
Error Term ◽  
Nous Obtenons ◽  
Threshold Phenomenon ◽  
Arbitrary Genus ◽  
International Audience ◽  
Asymptotic Probability ◽  
Number Of Cycles ◽  
Pour Cent ◽  
Orientable Surfaces

International audience A permutation $a_1a_2 \ldots a_n$ is $\textit{indecomposable}$ if there does not exist $p \lt n$ such that $a_1a_2 \ldots a_p$ is a permutation of $\{ 1,2, \ldots ,p\}$. We compute the asymptotic probability that a permutation of $\mathbb{S}_n$ with $m$ cycles is indecomposable as $n$ goes to infinity with $m/n$ fixed. The error term is $O(\frac{\log(n-m)}{ n-m})$. The asymptotic probability is monotone in $m/n$, and there is no threshold phenomenon: it degrades gracefully from $1$ to $0$. When $n=2m$, a slight majority ($51.1 \ldots$ percent) of the permutations are indecomposable. We also consider indecomposable fixed point free involutions which are in bijection with maps of arbitrary genus on orientable surfaces, for these involutions with $m$ left-to-right maxima we obtain a lower bound for the probability of being indecomposable. Une permutation $a_1a_2 \ldots a_n$ est $\textit{indécomposable}$, s’il n’existe pas de $p \lt n$ tel que $a_1a_2 \ldots a_p$ est une permutation de $\{ 1,2, \ldots ,p\}$. Nous calculons la probabilité pour qu’une permutation de $\mathbb{S}_n$ ayant $m$ cycles soit indécomposable et plus particulièrement son comportement asymptotique lorsque $n$ tend vers l’infini et que $m=n$ est fixé. Cette valeur décroît régulièrement de $1$ à $0$ lorsque $m=n$ croît, et il n’y a pas de phénomène de seuil. Lorsque $n=2m$, une faible majorité ($51.1 \ldots$ pour cent) des permutations sont indécomposables. Nous considérons aussi les involutions sans point fixe indécomposables qui sont en bijection avec les cartes de genre quelconque plongées dans une surface orientable, pour ces involutions ayant $m$ maxima partiels (ou records) nous obtenons une borne inférieure pour leur probabilité d’êtres indécomposables.

scholarly journals A Short Approach to Catalan Numbers Modulo $2^r$

10.37236/664 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Guoce Xin ◽  
Jing-Feng Xu
Keyword(s):  
Catalan Numbers ◽  
Binary Expansion ◽  
Dyck Paths ◽  
Combinatorial Interpretations ◽  
Recent Classification ◽  
Motzkin Paths

We notice that two combinatorial interpretations of the well-known Catalan numbers $C_n=(2n)!/n!(n+1)!$ naturally give rise to a recursion for $C_n$. This recursion is ideal for the study of the congruences of $C_n$ modulo $2^r$, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The equivalence $C_{n}\equiv_{2^r} C_{\bar n}$ is further reduced to $C_{n}\equiv_{2^r} C_{\tilde{n}}$ for simpler $\tilde{n}$. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we find new classes of combinatorial sequences whose $2$-adic order is equal to that of $C_n$, which is one less than the sum of the digits of the binary expansion of $n+1$.

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