New Wilf-equivalence results for vincular patterns

Kasraoui, Anisse

doi:10.1016/j.ejc.2012.07.006

Cited by 8 publications

(16 citation statements)

References 17 publications

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“…Finally, we thank Andrew Baxter for clarifying the status of the conjecture about 14 23-avoiding permutations, which brought reference [22] to our attention.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…arXiv:1702.04529v3 [math.CO] 11 Jan 2018⊆ Av(2 41 3, 3 41 2) ⊆ Figure 1: Sequences from Catalan to factorial numbers, with nested families of pattern-avoiding permutations that they enumerate.The focus of this paper is the study of the two sequences of semi-Baxter and strong-Baxter numbers.We deal with the semi-Baxter sequence (enumerating semi-Baxter permutations) in Section 3. It has been proved in [22] (as a special case of a general statement) that this sequence also enumerates plane permutations, defined by the avoidance of 2 14 3. This sequence is referenced as A117106 in [23].…”

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confidence: 95%

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Semi-Baxter and Strong-Baxter: Two Relatives of the Baxter Sequence

Bouvel¹,

Guerrini²,

Rechnitzer³

et al. 2018

SIAM J. Discrete Math.

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In this paper, we enumerate two families of pattern-avoiding permutations: those avoiding the vincular pattern 2 41 3, which we call semi-Baxter permutations, and those avoiding the vincular patterns 2 41 3, 3 14 2 and 3 41 2, which we call strong-Baxter permutations. We call semi-Baxter numbers and strong-Baxter numbers the associated enumeration sequences. We prove that the semi-Baxter numbers enumerate in addition plane permutations (avoiding 2 14 3). The problem of counting these permutations was open and has given rise to several conjectures, which we also prove in this paper.For each family (that of semi-Baxter -or equivalently, plane -and that of strong-Baxter permutations), we describe a generating tree, which translates into a functional equation for the generating function. For semi-Baxter permutations, it is solved using (a variant of) the kernel method: this gives an expression for the generating function while also proving its D-finiteness. From the obtained generating function, we derive closed formulas for the semi-Baxter numbers, a recurrence that they satisfy, as well as their asymptotic behavior. For strong-Baxter permutations, we show that their generating function is (a slight modification of) that of a family of walks in the quarter plane, which is known to be non D-finite. arXiv:1702.04529v3 [math.CO] 11 Jan 2018⊆ Av(2 41 3, 3 41 2) ⊆ Figure 1: Sequences from Catalan to factorial numbers, with nested families of pattern-avoiding permutations that they enumerate.The focus of this paper is the study of the two sequences of semi-Baxter and strong-Baxter numbers.We deal with the semi-Baxter sequence (enumerating semi-Baxter permutations) in Section 3. It has been proved in [22] (as a special case of a general statement) that this sequence also enumerates plane permutations, defined by the avoidance of 2 14 3. This sequence is referenced as A117106 in [23]. We first give a more specific proof that plane permutations and semi-Baxter permutations are equinumerous, by providing a common generating tree (or succession rule) with two labels for these two families. Basics and references about generating trees can be found in Section 2.We solve completely the problem of enumerating semi-Baxter permutations (or equivalently, plane permutations), pushing further the techniques that were used to enumerate Baxter permutations in [10]. Namely, we start from the functional equation associated with our succession rule for semi-Baxter permutations, and we solve it using variants of the kernel method [10,21]. This results in an expression for the generating function for semi-Baxter permutations, showing that this generating function is D-finite 2 . From it, we obtain several formulas for the semi-Baxter numbers: first, a complicated closed formula; second, a simple recursive formula; and third, three simple closed formulas that were conjectured by D. Bevan [7].The problem of enumerating plane permutations was posed by M. Bousquet-Mélou and S. Butler in [12]. Some conjectures related to this enumeration problem w...

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“…Finally, we thank Andrew Baxter for clarifying the status of the conjecture about 14 23-avoiding permutations, which brought reference [22] to our attention.…”

Section: Acknowledgementsmentioning

confidence: 99%

mentioning

confidence: 95%

Semi-Baxter and Strong-Baxter: Two Relatives of the Baxter Sequence

Bouvel¹,

Guerrini²,

Rechnitzer³

et al. 2018

SIAM J. Discrete Math.

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“…For example, the permutation 1234 has four occurrences of 1 2 3, but there is no permutation of length 4 with four occurrences of 1 3 2, Just as finding the distribution of occurrences of a pattern is in general harder than finding the number of avoiders, so there are still few results about strong Wilf equivalence. In fact, the only nontrivial such results I am aware of are recent results of Kasraoui [30], who gives an infinite family of strong Wilf equivalences for non-classical vincular patterns, including, for example, the equivalence of 3 421 and 421 3.…”

Section: -2-4-1mentioning

confidence: 99%

Some open problems on permutation patterns

Steingrı́msson

2013

Surveys in Combinatorics 2013

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This is a brief survey of some open problems on permutation patterns, with an emphasis on subjects not covered in the recent book by Kitaev, Patterns in Permutations and words. I first survey recent developments on the enumeration and asymptotics of the pattern 1324, the last pattern of length 4 whose asymptotic growth is unknown, and related issues such as upper bounds for the number of avoiders of any pattern of length k for any given k. Other subjects treated are the Möbius function, topological properties and other algebraic aspects of the poset of permutations, ordered by containment, and also the study of growth rates of permutation classes, which are containment closed subsets of this poset.2 In this section I write classical patterns with dashes between all pairs of adjacent letters, to distinguish them from other vincular and bivincular patterns. In later sections, I will write classical patterns in the classical way, without any dashes, to keep the notation less cumbersome.

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“…As a consequence of Theorem 2.1(iv), we will deduce the following result about permutation patterns, originally conjectured by Baxter and Pudwell [3,Conjecture 17], and later proved by Baxter and Shattuck [4,Corollary 11] and by Kasraoui [15,Corolary 1.9(a)]. Here we present a direct bijective proof based on consecutive patterns of relations in inversion sequences, which is simpler than the previously known proofs.…”

Section: Corollary 22 Wilf Equivalence and Strong Wilf Equivalence Cl...mentioning

confidence: 66%

Consecutive patterns in inversion sequences II: avoiding patterns of relations

Auli¹,

Elizalde²

2019

Preprint

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Inversion sequences are integer sequences e = e 1 e 2 . . . e n such that 0 ≤ e i < i for each i.The study of patterns in inversion sequences was initiated by Corteel-Martinez-Savage-Weselcouch and Mansour-Shattuck in the classical (non-consecutive) case, and later by Auli-Elizalde in the consecutive case, where the entries of a pattern are required to occur in adjacent positions. In this paper we continue this investigation by considering consecutive patterns of relations, in analogy to the work of Martinez-Savage in the classical case. Specifically, given two binary relations R 1 , R 2 ∈ {≤, ≥, <, >, =, =}, we study inversion sequences e with no subindex i such that e i R 1 e i+1 R 2 e i+2 .By enumerating such inversion sequences according to their length, we obtain well-known quantities such as Catalan numbers, Fibonacci numbers and central polynomial numbers, relating inversion sequences to other combinatorial structures. We also classify consecutive patterns of relations into Wilf equivalence classes, according to the number of inversion sequences avoiding them, and into more restrictive classes that consider the positions of the occurrences of the patterns.As a byproduct of our techniques, we obtain a simple bijective proof of a result of Baxter-Shattuck and Kasraoui about Wilf-equivalence of vincular patterns, and we prove a conjecture of Martinez and Savage, as well as related enumeration formulas for inversion sequences satisfying certain unimodality conditions.

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New Wilf-equivalence results for vincular patterns

Cited by 8 publications

References 17 publications

Semi-Baxter and Strong-Baxter: Two Relatives of the Baxter Sequence

Semi-Baxter and Strong-Baxter: Two Relatives of the Baxter Sequence

Some open problems on permutation patterns

Consecutive patterns in inversion sequences II: avoiding patterns of relations

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