Bijections for refined restricted permutations

Elizalde, Sergi; Pak, Igor

doi:10.1016/j.jcta.2003.10.009

Cited by 53 publications

(103 citation statements)

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“…It also appears in [21], as a bijection between 132-avoiding permutations and Dyck paths. Moreover, we would like to point out that a simple visualization of this bijection, which involves lattice paths connecting opposite corners of the permutation array, is given in [16,26]. Using such a description, some of the properties of the Bruhat order on noncrossing partitions, such as Theorem 4.2, can be suitably rephrased.…”

Section: Relationship With the Strong Bruhat Order On Permutationsmentioning

confidence: 99%

A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

Barcucci

Bernini

Ferrari³

et al. 2005

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In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135 (2005), 77-92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math. 217 (2000), 367-409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.

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Section: Relationship With the Strong Bruhat Order On Permutationsmentioning

confidence: 99%

A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

Barcucci

Bernini

Ferrari³

et al. 2005

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“…The longest increasing subsequence of a pattern-avoiding permutation was studied in [9] and the structure of the fixed points in pattern-avoiding permutations has been the subject many papers [12,10,11,13,27].…”

Section: Introductionmentioning

confidence: 99%

Pattern‐avoiding permutations and Brownian excursion part I: Shapes and fluctuations

Hoffman

Rizzolo

Slivken

2016

Random Struct Algorithms

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Abstract. Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random permutation avoiding a pattern of length 3 and their relations to Brownian excursion. Exploring this connection to Brownian excursion allows us to strengthen the recent results of Madras and Pehlivan [23] and Miner and Pak [27] as well as to understand many of the interesting phenomena that had previously gone unexplained.

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“…Elizalde and Pak in [3] prove the corresponding result for their bijection Θ. We now establish the analogue of Theorem 3 for the map φ * studied in [1] and [2].…”

Section: Theorem 4 γ Maps the Set Of Involutions In S N (321) Onto Tmentioning

confidence: 60%

“…Three decades later a refinement of this result was given by Robertson,Saracino,and Zeilberger [7], by taking into account the number of fixed points of σ . If S k n (α) denotes the set of σ ∈ S n (α) with exactly k fixed points, it was shown in [7] that |S k n (321)| = |S k n (132)| = |S k n (213)| and |S k n (231)| = |S k n (312)| for all 0 k n. The proofs in [7] were nonbijective, and Elizalde and Pak soon improved the result |S k n (321)| = |S k n (132)| by giving in [3] a bijection Θ : S n (321) → S n (132) that preserves not only the number of fixed points but also the number of excedances (i's such that σ i > i). The bijection Θ is obtained by combining two previously known bijections: a bijection of Knuth [4] between S n (321) and the set D n of Dyck paths of length 2n, and (in modified form) a bijection of Krattenthaler [5] between S n (132) and D n .…”

Section: Introductionmentioning

confidence: 99%

On bijections for pattern-avoiding permutations

Bloom¹,

Saracino

2009

Journal of Combinatorial Theory, Series A

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By considering bijections from the set of Dyck paths of length 2n onto each of S n (321) and S n (132), Elizalde and Pak in [S. Elizalde, I. Pak, Bijections for refined restricted permutations, J. Combin. Theory Ser. A 105 (2004) 207-219] gave a bijection Θ : S n (321) → S n (132) that preserves the number of fixed points and the number of excedances in each σ ∈ S n (321). We show that a direct bijection Γ : S n (321) → S n (132) introduced by Robertson in [A. Robertson, Restricted permutations from Catalan to Fine and back, Sém. Lothar. Combin. 50 (2004) B50g] also preserves the number of fixed points and the number of excedances in each σ . We also show that a bijection φ * : S n (213) → S n (321) studied in [J. Backelin, J. West, G. Xin, Wilf-equivalence for singleton classes, Adv. in Appl. Math. 38 (2007) 133-148] and [M. Bousquet-Melou, E. Steingrimsson, Decreasing subsequences in permutations and Wilf equivalence for involutions, J. Algebraic Combin. 22 (2005) 383-409] preserves these same statistics, and we show that an analogous bijection from S n (132) onto S n (213) does the same.

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Bijections for refined restricted permutations

Cited by 53 publications

References 10 publications

A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

Pattern‐avoiding permutations and Brownian excursion part I: Shapes and fluctuations

On bijections for pattern-avoiding permutations

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