On Wilf Equivalence for Alternating Permutations

Sherry H.F. Yan

doi:10.37236/3243

On Wilf Equivalence for Alternating Permutations

The Electronic Journal of Combinatorics ◽

10.37236/3243 ◽

2013 ◽

Vol 20 (3) ◽

Cited By ~ 2

Author(s):

Sherry H.F. Yan

Keyword(s):

Alternating Permutations ◽

Wilf Equivalence

In this paper, we obtain several new classes of Wilf-equivalent patterns for alternating permutations. In particular, we prove that for any nonempty pattern $\tau$, the patterns $12\ldots k\oplus\tau$ and $k\ldots 21\oplus\tau$ are Wilf-equivalent for alternating permutations, paralleling a result of Backelin, West, and Xin for Wilf equivalence for permutations.

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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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Ascent-Descent Young Diagrams and Pattern Avoidance in Alternating Permutations

The Electronic Journal of Combinatorics ◽

10.37236/3244 ◽

2014 ◽

Vol 21 (3) ◽

Author(s):

Ravi Jagadeesan

Keyword(s):

Pattern Avoidance ◽

Electronic Journal ◽

Young Diagrams ◽

Alternating Permutations ◽

Wilf Equivalence

We investigate pattern avoidance in alternating permutations and an alternating analogue of Young diagrams. In particular, using an extension of Babson and West's notion of shape-Wilf equivalence described in our recent paper (with N. Gowravaram), we generalize results of Backelin, West, and Xin and Ouchterlony to alternating permutations. Unlike Ouchterlony and Bóna's bijections, our bijections are not the restrictions of Backelin, West, and Xin's bijections to alternating permutations. This paper is the second of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (with N. Gowravaram, arXiv:1301.6796v1). The first paper in the series is Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (with N. Gowravaram, Electronic Journal of Combinatorics 20(4):#P17, 2013).

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On a Family of Conjectures of Joel Lewis on Alternating Permutations

Graphs and Combinatorics ◽

10.1007/s00373-013-1291-2 ◽

2013 ◽

Vol 30 (3) ◽

pp. 521-526 ◽

Cited By ~ 1

Author(s):

Miklós Bóna

Keyword(s):

Alternating Permutations

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A Note on Alternating Permutations

American Mathematical Monthly ◽

10.1080/00029890.2007.11920432 ◽

2007 ◽

Vol 114 (5) ◽

pp. 437-440 ◽

Cited By ~ 1

Author(s):

Anthony Mendes

Keyword(s):

Alternating Permutations

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q-Enumeration of alternating permutations of odd length

Journal of Discrete Mathematical Sciences and Cryptography ◽

10.1080/09720529.2010.10698276 ◽

2010 ◽

Vol 13 (1) ◽

pp. 45-67

Author(s):

Matodzi S. Mulaudzi ◽

O. Akinyemi

Keyword(s):

Alternating Permutations

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A General Theory of Wilf-Equivalence for Catalan Structures

The Electronic Journal of Combinatorics ◽

10.37236/5629 ◽

2015 ◽

Vol 22 (4) ◽

Cited By ~ 3

Author(s):

Michael Albert ◽

Mathilde Bouvel

Keyword(s):

General Theory ◽

Equivalence Relation ◽

Generating Functions ◽

Asymptotic Estimate ◽

Equivalence Classes ◽

Catalan Number ◽

Combinatorial Structures ◽

Dyck Paths ◽

Significant Interest ◽

Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

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