Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one‐dimensional labels

Borga, Jacopo

doi:10.1002/rsa.21005

Cited by 10 publications

(9 citation statements)

References 48 publications

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“…In particular, we believe that a good choice would be a set of (possibly generalized) patterns B for which the corresponding family Av(B) have been enumerated through generating trees. Indeed, the first author of this article has recently shown in [Bor20a] that generating trees behave well in the analysis of consecutive patterns of permutations in these families. We believe that generating trees would be particularly helpful to prove some analogues of Lemma 4.14 -that is the key lemma in the proof of Theorem 1.11 -for other families of permutations.…”

Section: Future Projects and Open Questionsmentioning

confidence: 95%

The feasible regions for consecutive patterns of pattern-avoiding permutations

Borga¹,

Penaguiao²

2020

Preprint

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We study the feasible region for consecutive patterns of pattern-avoiding permutations. More precisely, given a family C of permutations avoiding a fixed set of patterns, we study the limit of proportions of consecutive patterns on large permutations of C. These limits form a region, which we call the pattern-avoiding feasible region for C. We show that, when C is the family of τ -avoiding permutations, with either τ of size three or τ a monotone pattern, the pattern-avoiding feasible region for C is a polytope. We also determine its dimension using a new tool for the monotone pattern case whereby we are able to compute the dimension of the image of a polytope after a projection.We further show some general results for the pattern-avoiding feasible region for any family C of permutations avoiding a fixed set of patterns, and we conjecture a general formula for its dimension.Along the way, we discuss connections of this work with the problem of packing patterns in pattern-avoiding permutations and to the study of local limits for pattern-avoiding permutations.

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Section: Future Projects and Open Questionsmentioning

confidence: 95%

The feasible regions for consecutive patterns of pattern-avoiding permutations

Borga¹,

Penaguiao²

2020

Preprint

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“…Moreover, we would like to explore several new families of permutations where a bijection with two-dimensional walks is available. A technique is presented in a recent work of the first author [Bor20a] to sample uniform permutations in families enumerated through generating trees as conditioned random colored walks. In [Bor20a], the analysis for determining a central limit theorem for proportions of consecutive patterns is restricted to one-dimensional walks, i.e.…”

Section: Generality Of Our Techniques and Open Problemsmentioning

confidence: 99%

“…A technique is presented in a recent work of the first author [Bor20a] to sample uniform permutations in families enumerated through generating trees as conditioned random colored walks. In [Bor20a], the analysis for determining a central limit theorem for proportions of consecutive patterns is restricted to one-dimensional walks, i.e. to families having a generating tree with one-dimensional labels.…”

Section: Generality Of Our Techniques and Open Problemsmentioning

confidence: 99%

Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

Borga¹,

Maazoun²

2020

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Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant, called tandem walks, are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes and we relate it to the three families mentioned above.We prove joint Benjamini-Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new random measure of the unit square, called the Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. On top of that, we relate the limiting objects of the four families to each other, both in the local and scaling limit case.The scaling limit result is based on the convergence of the trajectories of the coalescentwalk process to the coalescing flow -in the terminology of Le Jan and Raimond ( 2004)of a perturbed version of the Tanaka stochastic differential equation. Our scaling result entails joint convergence of the tandem walks of a plane bipolar orientation and its dual, extending the main result of Gwynne, Holden, Sun (2016), and answering more precisely Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (2019).

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“…Several previous works establish asymptotic normality of this distribution for different sets of patterns in permutations selected uniformly at random. For example, see Feller [25, 3rd ed., p.257] (for inversions), Mann [43] (for descents), Fulman [29] (for both inversions and descents), Goldstein [31] and Borga [11] (for consecutive patterns), Bóna [8] (for classical patterns) and Hofer [32] (for vincular patterns). However, the number of occurrences of some simple bivincular patterns is not normally distributed (see Section 6.3).…”

Section: Introductionmentioning

confidence: 99%

Moments of permutation statistics and central limit theorems

Dimitrov,

Khare

2021

Preprint

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We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns, which uses a lemma of Burstein and Hästö. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns. Closed forms for some of the higher moments of several popular statistics on permutations are also obtained.

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Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one‐dimensional labels

Cited by 10 publications

References 48 publications

The feasible regions for consecutive patterns of pattern-avoiding permutations

The feasible regions for consecutive patterns of pattern-avoiding permutations

Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

Moments of permutation statistics and central limit theorems

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