Pattern Avoidance Research Articles

Baxter permuton and Liouville quantum gravity

The Baxter permuton is a random probability measure on the unit square which describes the scaling limit of uniform Baxter permutations. We determine an explicit formula for the density of the expectation of the Baxter permuton. This answers a question of Dokos and Pak (Online J Anal Comb 9:12, 2014). We also prove that all pattern densities of the Baxter permuton are strictly positive, distinguishing it from other permutons arising as scaling limits of pattern-avoiding permutations. Our proofs rely on a recent connection between the Baxter permuton and Liouville quantum gravity (LQG) coupled with the Schramm-Loewner evolution (SLE). The method works equally well for a two-parameter generalization of the Baxter permuton recently introduced by the first author, except that the density is not as explicit. This new family of permutons, called skew Brownian permuton, describes the scaling limit of a number of random constrained permutations. We finally observe that in the LQG/SLE framework, the expected proportion of inversions in a skew Brownian permuton equals $$\frac{\pi -2\theta }{2\pi }$$ where $$\theta $$ is the so-called imaginary geometry angle between a certain pair of SLE curves.

On automorphisms of undirected Bruhat graphs

The undirected Bruhat graph \(\Gamma (u,v)\) has the elements of the Bruhat interval [u, v] as vertices, with edges given by multiplication by a reflection. Famously, \(\Gamma (e,v)\) is regular if and only if the Schubert variety \(X_v\) is smooth, and this condition on v is characterized by pattern avoidance. In this work, we classify when \(\Gamma (e,v)\) is vertex-transitive; surprisingly this class of permutations is also characterized by pattern avoidance and sits nicely between the classes of smooth permutations and self-dual permutations. This leads us to a general investigation of automorphisms of \(\Gamma (u,v)\) in the course of which we show that special matchings, which originally appeared in the theory of Kazhdan–Lusztig polynomials, can be characterized, for the symmetric and right-angled groups, as certain \(\Gamma (u,v)\)-automorphisms which are conjecturally sufficient to generate the orbit of e under \({{\,\textrm{Aut}\,}}(\Gamma (e,v))\).

Securing High-Resolution Images From Unmanned Aerial Vehicles With DNA Encoding and Bit-Plane Extraction Method

Unmanned aerial vehicles (UAVs) are getting more popular for deployment in surveillance related operations owing to their flexibility and ability to reach hazardous areas. Moreover, the quality of digital cameras is getting better, and they can capture and store more visual information in high-resolution images. Unfortunately, due to the resource-constrained nature of UAVs, storing such large images can exhaust memory and related resources; whereas, the transmission of these images over the public link can pose several security threats. Securing these critical images during transmission from unauthorized access can be achieved through the use of efficient encryption techniques. This article proposes a novel encryption scheme incorporating both confusion and diffusion for encrypting both grey-scale and color images. In the proposed- encryption scheme, the image blocks are rearranged using a combination of random permutation, rotation, DNA encoding and zigzag pattern. Next, a bit-plane extraction method is used to obtain eight different bit-planes, including the most and least significant ones, from the scrambled image. These extracted bit-planes are then processed using confusion and diffusion techniques with a secret key, which is created using a hyper-chaotic map. The proposed method for encrypting the images taken by unmanned ariel vehicles is evaluated by examining its security level and time complexity using evaluation metrics such as correlation, entropy, energy, histogram analysis, keyspace and key sensitivity. The results and analysis demonstrated that the proposed encryption algorithm is able to effectively secure digital images. Additionally, the proposed work is also found to be superior to existing methods when compared using statistical security metrics.

Symbolic convergent cross mapping based on permutation mutual information

In this paper, we extend convergent cross mapping (CCM) and propose symbolic CCM (SCCM), which uses mutual information based on permutation pattern instead of Pearson correlation coefficient to estimate cross-mapping ability. We numerically demonstrate that SCCM is a robust method for quantifying information flow between time series in chaotic systems, even under the influence of noises. Using the method, we analyze the multichannel EEG signals of ADHD children and control children, and identify the differences between the two groups of subjects with reliable results.

On a Conjecture About Strong Pattern Avoidance

On a Conjecture About Strong Pattern Avoidance

Length-Four Pattern Avoidance in Inversion Sequences

Inversion sequences of length $n$ are integer sequences $e_1,\ldots ,e_n$ with $0\le e_i<i$ for all $i$, which are in bijection with the permutations of length $n$. In this paper, we classify all Wilf equivalence classes of pattern-avoiding inversion sequences of length-4 patterns except for one case (whether 3012 $\equiv$ 3201) and enumerate some of the length-4 pattern-avoiding inversion sequences that are in the OEIS.

Lexical Verb Forms in L1 and L2 Spoken English: A Corpus-Based Analysis

The present study investigates Korean learners’ use of English lexical verb forms and identifies learner deviations from native usage patterns. It examines the frequencies of six verb forms (i.e., base, present, third-person singular, past, gerund, and past participle) in a native and a learner corpus of argumentative speeches. The study reveals that certain verb forms are overor underused by Korean learners of L2 English, compared to native speakers of English. It is also found that L2 overuse and underuse of verb forms are closely related to Korean learners’ overreliance on or avoidance of major lexico-grammatical patterns around the verb forms (e.g., to + base verb). These findings have pedagogical implications for teaching Korean learners to use various verb forms with major lexico-grammatical patterns in English speaking.

Permutation Lempel-Ziv 복잡성 척도를 적용한 근 피로도 추정

A new approach for quantifying muscle fatigue is presented by analyzing the deterministic complexity in the surface electromyogram(SEMG) using Permutation Lempel-Ziv(PLZ) complexity measure. The PLZ complexity measure provide a metric for the number of distinct deterministic patterns in signals by combining permutation patterns and Lempel-Ziv complexity analysis. For muscle fatigue experiment by using PLZ complexity measure, surface EMG signals from eleven subjects were recorded in biceps brachii muscle with constant isometric 20, 50, and 80%MVC contractions. The PLZ complexity measure is evaluated with the performance of muscle fatigue estimation for the SEMG signals compared to the performance of previous reported LZ complexity measure. This PLZ complexity measure has advantages that is computationally easy to compute and does not require any assumptions about the stationarity of the SEMG signal.

Letter Graphs and Geometric Grid Classes of Permutations

We uncover a connection between two seemingly unrelated notions: lettericity, from structural graph theory, and geometric griddability, from the world of permutation patterns. Both of these notions capture important structural properties of their respective classes of objects. We prove that these notions are equivalent in the sense that a permutation class is geometrically griddable if and only if the corresponding class of inversion graphs has bounded lettericity.

Combinatorial Generation via Permutation Languages. III. Rectangulations

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants, including aspect-ratio-universal rectangulations. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework.

The feasible regions for consecutive patterns of pattern-avoiding permutations

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 consider the limit of proportions of consecutive patterns on large permutations of C. These limits form a region, which we call the consecutive patterns feasible region forC.We determine the dimension of the consecutive patterns feasible region for all families C closed either for the direct sum or the skew sum. These families include for instance the ones avoiding a single pattern and all substitution-closed classes. We further show that these regions are always convex and we conjecture that they are always polytopes. We prove this conjecture when C is the family of τ-avoiding permutations, with either τ of size three or τ a monotone pattern. Furthermore, in these cases we give a full description of the vertices of these polytopes via cycle polytopes.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.

Classical and consecutive pattern avoidance in rooted forests

Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern σ∈Sk if in each tree, each sequence of labels along the shortest path from the root to a vertex does not contain a subsequence with the same relative order as σ. For each permutation σ∈Sk−2, we construct a bijection between n-vertex forests avoiding (σ)(k−1)k≔σ(1)⋯σ(k−2)(k−1)k and n-vertex forests avoiding (σ)k(k−1)≔σ(1)⋯σ(k−2)k(k−1), giving a common generalization of results of West on permutations and Anders–Archer on forests. We further define a new object, the forest-Young diagram, which we use to extend the notion of shape-Wilf equivalence to forests. In particular, this allows us to generalize the above result to a bijection between forests avoiding {(σ1)k(k−1),(σ2)k(k−1),…,(σℓ)k(k−1)} and forests avoiding {(σ1)(k−1)k,(σ2)(k−1)k,…,(σℓ)(k−1)k} for σ1,…,σℓ∈Sk−2. Furthermore, we give recurrences enumerating the forests avoiding {123⋯k}, {213}, and other sets of patterns. Finally, we extend the Goulden–Jackson cluster method to study consecutive pattern avoidance in rooted trees as defined by Anders and Archer. Using the generalized cluster method, we prove that if two length-k patterns are strong-c-forest-Wilf equivalent, then up to complementation, the two patterns must start with the same number. We also prove the surprising result that the patterns 1324 and 1423 are strong-c-forest-Wilf equivalent, even though they are not c-Wilf equivalent with respect to permutations.

Strings from linear recurrences and permutations: A Gray code

Each positive increasing integer sequence {an}n≥0 can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of k-generalized Fibonacci sequences leading to the binary strings avoiding 1k. We prove a bijection between the set of such strings of length n and the set Sn+1(321,312,23…(k+1)1) which is a subset of the symmetric group Sn+1 of the permutations of length n+1 avoiding the permutation patterns 321,312, and 234…(k+1)1, where 234…(k+1) stands for the sequence of increasing integers from 2 to k+1.Finally, basing on a known Gray code for those strings, we define a Gray code for Sn+1(321,312,234…(k+1)1), where two consecutive permutations differ by an adjacent transposition.

Labelled well-quasi-order for permutation classes

While the theory of labelled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and show how labelled well quasi order provides a lens through which to view and extend previous well-quasi-order results in the permutation patterns literature. Connections to the graph setting are emphasised throughout. In particular, we establish that a permutation class is labelled well-quasi-ordered if and only if its corresponding graph class is also labelled well-quasi-ordered.Mathematics Subject Classifications: 05A05, 06A07Keywords: Labelled well-quasi-order, permutation patterns, well-quasi-order

Positional Marked Patterns in Permutations

We define and study positional marked patterns, permutations $\tau$ where one of elements in $\tau$ is underlined. Given a permutation $\sigma$, we say that $\sigma$ has a $\tau$-match at position $i$ if $\tau$ occurs in $\sigma$ in such a way that $\sigma_i$ plays the role of the underlined element in the occurrence. We let $pmp_\tau(\sigma)$ denote the number of positions $i$ which $\sigma$ has a $\tau$-match. This defines a new class of statistics on permutations, where we study such statistics and prove a number of results. In particular, we prove that two positional marked patterns $1\underline{2}3$ and $1\underline{3}2$ give rise to two statistics that have the same distribution. The equidistibution phenomenon also occurs in other several collections of patterns like $\left \{1\underline{2}3 , 1\underline{3}2 \right \}$, and $\left \{ 1\underline234, 1\underline243, \underline2134, \underline2 1 4 3 \right \}$, as well as two positional marked patterns of any length $n$: $\left \{ 1\underline 2\tau , \underline 21\tau \right \}$.

Classical Length-5 Pattern-Avoiding Permutations

We have made a systematic numerical study of the 16 Wilf classes of length-5 classical pattern-avoiding permutations from their generating function coefficients. We have extended the number of known coefficients in fourteen of the sixteen classes. Careful analysis, including sequence extension, has allowed us to estimate the growth constant of all classes, and in some cases to estimate the sub-dominant power-law term associated with the exponential growth.
 In six of the sixteen classes we find the familiar power-law behaviour, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot n^g,$ while in the remaining ten cases we find a stretched exponential as the most likely sub-dominant term, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot \mu_1^{n^\sigma} \cdot n^g,$ where $0 < \sigma < 1.$ We have also classified the 120 possible permutations into the 16 distinct classes.
 We give compelling numerical evidence, and in one case a proof, that all 16 Wilf-class generating function coefficients can be represented as moments of a non-negative measure on $[0,\infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as log-convexity, which can be used to provide quite strong rigorous lower bounds.
 Stronger bounds still can be established under plausible monotonicity assumptions about the terms in the continued-fraction expansion of the generating functions implied by the Stieltjes property. In this way we provide strong (non-rigorous) lower bounds to the growth constants, which are sometimes within a few percent of the exact value.

Dispersion entropy-based Lempel-Ziv complexity: A new metric for signal analysis

Lempel-Ziv complexity (LZC) is one of the most important metrics for detecting dynamic changes in non-linear signals, but due to its dependence on binary conversion, LZC tends to lose some of the effective information of the time series, while the noise immunity is not guaranteed and cannot be applied to the detection of real-world signals. To address these limitations, we have developed a dispersion entropy-based LZC (DELZC) based on the normal cumulative distribution function (NCDF) and dispersion permutation patterns. In DELZC, the time series are first processed by NCDF to increase the number of classes and thus reduce the loss of information, and in addition, the dispersive entropy (DE) in terms of the ordinal number of the permutation pattern is considered to replace the binary conversion of LZC, thus improving the ability to capture the dynamic changes in the time series. In signal analysis using a set of time series, several easy-to-understand concepts are used to demonstrate the superiority of DELZC over other three LZC metrics in detecting the dynamic variability of the signals, namely LZC, dispersion LZC (DLZC) and permutation LZC (PLZC). The synthetic signal experiments demonstrate the superiority of DELZC in detecting the dynamic changes of time series and characterizing the complexity of signal, and also have lower noise sensitivity. Moreover, DELZC has the best performance in diagnosing four states of rolling bearing fault signals and classifying five types of ship radiation noise signals, with higher recognition rates than LZC, PLZC and DLZC.

Subregularity in infinitely labeled generating trees of restricted permutations

In this paper, we revisit the application of generating trees to the pattern avoidance problem for permutations. In particular, we study this problem for certain general sets of patterns and propose a new procedure leveraging the FinLabel algorithm and exploiting the subregularities in the associated generating trees. We consider some general kinds of generating trees for which the FinLabel algorithm fails to determine in a finite number of iterations the generating function that enumerates the underlying class of permutations. Our procedure provides a unified approach in these cases leading to a system of equations satisfied by a certain finite set of generating functions which can be readily solved with the aid of programming.

The features and geometry of tone in Laal

AbstractFeatures are standard in segmental analysis but have been less successfully applied to tone. Subtonal features have even been argued to be less satisfactory for the representation of African tone than tonal primitives such as H, M, L (Hyman 2010; Clements et al.2010). I argue that the two-feature system of Yip (1980) and Pulleyblank (1986) offers a straightforward account of the tonology of Laal, an endangered, three-tone isolate of southern Chad – in particular properties of the Mid tone that are otherwise difficult to account for, namely the avoidance of complex patterns involving M, and a pervasive M-to-L lowering process, both straightforwardly analysed as subtonal assimilation. Other tonal operations in Laal are shown to involve full-tone behaviour, justifying a tone geometry à la Snider (1999, 2020) where subtonal features are linked to a Tonal Root Node, giving tones the ability to be either fully or partially active, just like segments.

On avoiding 1233

In this paper, we establish a recurrence relation for finding the generating function for the number of k-ary words of length n that avoid 1233 for arbitrary k. Comparable generating function formulas may also be found counting words where a single permutation pattern of length three is avoided in addition to 1233.