scholarly journals Matchings Avoiding Partial Patterns

10.37236/1138 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
William Y. C. Chen ◽  
Toufik Mansour ◽  
Sherry H. F. Yan
Keyword(s):  
Catalan Numbers ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
Generating Trees ◽  
Super Catalan Numbers ◽  
Wilf Equivalence

We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers. We give a characterization of $12312$-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns $12312$ and $121323$ and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is derived by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we use the method of generating trees to show that the patterns 12132, 12123, 12321, 12231, 12213 are all equivalent to the pattern $12312$.

scholarly journals Chung-Feller Property of Schröder Objects

10.37236/5659 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Youngja Park ◽  
Sangwook Kim
Keyword(s):  
Connected Components ◽  
Combinatorial Interpretation ◽  
Noncrossing Partitions ◽  
Alternating Sign Matrices ◽  
Feller Property ◽  
Bijective Proof ◽  
Schröder Numbers ◽  
Schröder Paths

Large Schröder paths, sparse noncrossing partitions, partial horizontal strips, and $132$-avoiding alternating sign matrices are objects enumerated by Schröder numbers. In this paper we give formula for the number of Schröder objects with given type and number of connected components. The proofs are bijective using Chung-Feller style. A bijective proof for the number of Schröder objects with given type is provided. We also give a combinatorial interpretation for the number of small Schröder paths.

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 Generating trees and the Catalan and Schröder numbers

1995 ◽  
Vol 146 (1-3) ◽  
pp. 247-262 ◽  
Author(s):  
Julian West
Keyword(s):  
Schröder Numbers ◽  
Generating Trees

scholarly journals Remarks on Catalan and super-Catalan numbers

2011 ◽  
Vol 96 ◽  
pp. 237-244
Author(s):  
Anna Dorota Krystek ◽  
Łukasz Jan Wojakowski
Keyword(s):  
Catalan Numbers ◽  
Super Catalan Numbers

scholarly journals On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers

10.37236/2313 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Kendra Killpatrick
Keyword(s):  
Equivalence Classes ◽  
Catalan Numbers ◽  
Major Index ◽  
Powers Of 2 ◽  
Set Of Permutations ◽  
Adic Valuation ◽  
Permutation Statistic ◽  
Wilf Equivalence

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan.  In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \geq 0$.  The polynomial $F_n^{ch}(321;q) = \sum_{\sigma \in Av_n(321)} q^{ch(\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$.  We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \geq 1$, giving a refinement of the "if" direction of the $C_n$ parity result.  Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index.    In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\Pi$ and $\Pi'$.  We say $\Pi$ and $\Pi'$ are $st$-Wilf equivalent if $\sum_{\sigma \in Av_n(\Pi)} q^{st(\sigma)} = \sum_{\sigma \in Av_n(\Pi')} q^{st(\sigma)}$.  In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.

scholarly journals Congruences on Sums of Super Catalan Numbers

2018 ◽  
Vol 73 (4) ◽  
Author(s):  
Ji-Cai Liu
Keyword(s):  
Catalan Numbers ◽  
Super Catalan Numbers

The m-Schröder paths and m-Schröder numbers

2021 ◽  
Vol 344 (2) ◽  
pp. 112209
Author(s):  
Sheng-Liang Yang ◽  
Mei-yang Jiang
Keyword(s):  
Schröder Numbers ◽  
Schröder Paths

scholarly journals Generalized Small Schröder Numbers

10.37236/4827 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
JiSun Huh ◽  
SeungKyung Park
Keyword(s):  
Lattice Path ◽  
Dyck Paths ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
Positive Integers

We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from $(0,0)$ to $(2n,0)$ with the step set of  $\{(k,k), (l,-l), (2r,0)\, |\, k,l,r \in {\bf P}\}$, where ${\bf P}$ is the set of positive integers, which never goes below the $x$-axis, and with no horizontal steps at level 0.  We find a bijection between 5-colored Dyck paths and generalized small Schröder paths, proving that the number of generalized small Schröder paths is equal to $\sum_{k=1}^{n} N(n,k)5^{n-k}$ for $n\geq 1$.

scholarly journals Bijective Recurrences concerning Schröder Paths

10.37236/1385 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Robert A. Sulanke
Keyword(s):  
Lattice Paths ◽  
Schröder Numbers ◽  
Schröder Paths

Consider lattice paths in Z$^2$ with three step types: the up diagonal $(1,1)$, the down diagonal $(1,-1)$, and the double horizontal $(2,0)$. For $n \geq 1$, let $S_n$ denote the set of such paths running from $(0,0)$ to $(2n,0)$ and remaining strictly above the x-axis except initially and terminally. It is well known that the cardinalities, $r_n = |S_n|$, are the large Schröder numbers. We use lattice paths to interpret bijectively the recurrence $ (n+1) r_{n+1} = 3(2n - 1) r_{n} - (n-2) r_{n-1}$, for $n \geq 2$, with $r_1=1$ and $r_2=2$. We then use the bijective scheme to prove a result of Kreweras that the sum of the areas of the regions lying under the paths of $S_n$ and above the x-axis, denoted by $AS_n$, satisfies $ AS_{n+1} = 6 AS_n - AS_{n-1}, $ for $n \geq 2$, with $AS_1 =1$, and $AS_2 =7$. Hence $AS_n = 1, 7, 41, 239 ,1393, \ldots$. The bijective scheme yields analogous recurrences for elevated Catalan paths.

scholarly journals Enumeration of permutations sorted with two passes through a stack and D_8 symmetries

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Mathilde Bouvel ◽  
Olivier Guibert
Keyword(s):  
Generating Trees ◽  
International Audience

International audience We examine the sets of permutations that are sorted by two passes through a stack with a $D_8$ operation performed in between. From a characterization of these in terms of generalized excluded patterns, we prove two conjectures on their enumeration, that can be refined with the distribution of some statistics. The results are obtained by generating trees. On étudie les ensembles de permutations qui sont triées par deux passages dans une pile séparés par une opération du groupe $D_8$. À partir d'une caractérisation de ces ensembles en termes de motifs exclus généralisés, on démontre deux conjectures sur leur énumération, qui peuvent être raffinées par la distribution de certaines statistiques. Ces résultats sont obtenus à l'aide d'arbres de génération.

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