Generating Trees and Pattern Avoidance in Alternating Permutations

We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern $2143$. We use a generating tree approach to construct a recursive bijection between the set $A_{2n}(2143)$ of alternating permutations of length $2n$ avoiding $2143$ and the set of standard Young tableaux of shape $\langle n, n, n\rangle$, and between the set $A_{2n + 1}(2143)$ of alternating permutations of length $2n + 1$ avoiding $2143$ and the set of shifted standard Young tableaux of shape $\langle n + 2, n + 1, n\rangle$. We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.

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Pattern avoidance in alternating permutations and tableaux (extended abstract)

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2880 ◽

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 $312$.

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Generating trees for partitions and permutations with no k-nestings

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3050 ◽

2012 ◽

Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽

Author(s):

Sophie Burrill ◽

Sergi Elizalde ◽

Marni Mishna ◽

Lily Yen

Keyword(s):

Generating Functions ◽

Functional Equations ◽

Young Tableaux ◽

Set Partitions ◽

Lattice Walks ◽

Generating Trees ◽

Generating Tree ◽

International Audience ◽

Tree Approach

International audience We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter. Nous décrivons une approche, basée sur l'utilisation d'arbres de génération, pour énumération et la génération exhaustive de partitions et permutations sans k-emboîtement. Contrairement aux travaux antérieurs qui reposent sur un lien entre ces objets, tableaux de Young et familles de chemins dans des treillis, notre approche traite directement partitions et diagrammes de permutations. Nous fournissons des équations fonctionnelles explicites pour les séries génératrices, avec k en tant que paramètre.

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Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs

The Electronic Journal of Combinatorics ◽

10.37236/859 ◽

2008 ◽

Vol 15 (1) ◽

Author(s):

Marcos Kiwi ◽

Martin Loebl

Keyword(s):

Generating Function ◽

Maximum Size ◽

Pattern Avoidance ◽

Combinatorial Identities ◽

Young Tableaux ◽

Color Class ◽

Lattice Walks ◽

Type Property ◽

Bivariate Generating Function ◽

Standard Young Tableaux

We address the following question: When a randomly chosen regular bipartite multi–graph is drawn in the plane in the "standard way", what is the distribution of its maximum size planar matching (set of non–crossing disjoint edges) and maximum size planar subgraph (set of non–crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of $r$-regular bipartite multi–graphs with maximum planar matching (maximum planar subgraph) of at most $d$ edges to a signed sum of restricted lattice walks in ${\Bbb Z}^d$, and to the number of pairs of standard Young tableaux of the same shape and with a "descend–type" property. Our results are derived via generalizations of two combinatorial proofs through which Gessel's identity can be obtained (an identity that is crucial in the derivation of a bivariate generating function associated to the distribution of the length of LISs, and key to the analytic attack on Ulam's problem). Finally, we generalize Gessel's identity. This enables us to count, for small values of $d$ and $r$, the number of $r$-regular bipartite multi-graphs on $n$ nodes per color class with maximum planar matchings of size $d$.Our work can also be viewed as a first step in the study of pattern avoidance in ordered bipartite multi-graphs.

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Beyond Alternating Permutations: Pattern Avoidance in Young Diagrams and Tableaux

The Electronic Journal of Combinatorics ◽

10.37236/3246 ◽

2013 ◽

Vol 20 (4) ◽

Author(s):

Nihal Gowravaram ◽

Ravi Jagadeesan

Keyword(s):

Pattern Avoidance ◽

General Context ◽

Young Tableaux ◽

Young Diagrams ◽

Alternating Permutations ◽

Equivalence Type ◽

Shape Equivalence ◽

Pattern Avoiding Permutations ◽

Wilf Equivalence

We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young tableaux, that is, permutations that have $k-1$ ascents followed by a descent, followed by $k-1$ ascents, et cetera. We determine the relative sizes of the sets of pattern-avoiding $(k-1)$-ascent permutations in terms of the forbidden pattern. Furthermore, inequalities in the sizes of sets of pattern-avoiding permutations in this context arise from further extensions of shape-equivalence type enumerations. This paper is the first of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (arXiv:1301.6796v1). The second in the series is Ascent-descent Young diagrams and pattern avoidance in alternating permutations (by the second author, submitted).

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Alternating Permutations with Restrictions and Standard Young Tableaux

The Electronic Journal of Combinatorics ◽

10.37236/2306 ◽

2012 ◽

Vol 19 (2) ◽

Cited By ~ 2

Author(s):

Yuexiao Xu ◽

Sherry H. F. Yan

Keyword(s):

Intermediate Structure ◽

Young Tableaux ◽

Alternating Permutations ◽

One To One ◽

Standard Young Tableaux

In this paper, we establish bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating permutations of length $2n-1$ and the set of shifted standard Young tableaux of shape $(n+1, n, n-1)$ via an intermediate structure of Yamanouchi words. Moreover, we show that 4123-avoiding up-down alternating permutations of length $2n+1$ are in one-to-one correspondence with standard Young tableaux of shape $(n+1,n,n-1)$, and 4123-avoiding up-down alternating permutations of length $2n$ are in bijection with shifted standard Young tableaux of shape $(n+2,n,n-2)$.

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Subsequence Containment by Involutions

The Electronic Journal of Combinatorics ◽

10.37236/1911 ◽

2005 ◽

Vol 12 (1) ◽

Cited By ~ 2

Author(s):

Aaron D. Jaggard

Keyword(s):

Pattern Avoidance ◽

Young Tableaux ◽

Standard Young Tableaux

Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in ${\cal S}_{n}$ which contain a given permutation $\tau\in{\cal S}_{k}$ as a subsequence; this number depends on the patterns of the first $j$ values of $\tau$ for $1\leq j\leq k$. We then use this to define a partition of ${\cal S}_{k}$, analogous to Wilf-classes in the study of pattern avoidance, and examine properties of this equivalence. In the process, we show that a permutation $\tau_1\ldots\tau_k$ is layered iff, for $1\leq j\leq k$, the pattern of $\tau_1\ldots\tau_j$ is an involution. We also obtain a result of Sagan and Stanley counting the standard Young tableaux of size $n$ which contain a fixed tableau of size $k$ as a subtableau.

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Pattern Avoidance and Young Tableaux

The Electronic Journal of Combinatorics ◽

10.37236/6427 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 2

Author(s):

Zhousheng Mei ◽

Suijie Wang

Keyword(s):

General Class ◽

Pattern Avoidance ◽

Catalan Numbers ◽

Young Tableaux ◽

Standard Young Tableaux

This paper extends Lewis's bijection (J. Combin. Theorey Ser. A 118, 2011) to a bijection between a more general class $\mathcal{L}(n,k,I)$ of permutations and the set of standard Young tableaux of shape $\langle (k+1)^n\rangle$, so the cardinality\[|\mathcal{L}(n,k,I)|=f^{\langle (k+1)^n\rangle},\]is independent of the choice of $I\subseteq [n]$. As a consequence, we obtain some new combinatorial realizations and identities on Catalan numbers. In the end, we raise a problem on finding a bijection between $\mathcal{L}(n,k,I)$ and $\mathcal{L}(n,k,I')$ for distinct $I$ and $I'$.

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