Pattern Avoidance Research Articles

Coded Permutation Entropy: A Measure for Dynamical Changes Based on the Secondary Partitioning of Amplitude Information.

An improved permutation entropy (PE) algorithm named coded permutation entropy (CPE) is proposed in this paper to optimize the problems existing in PE based on the secondary partitioning. The principle of CPE algorithm is given, and the performance of it for dynamical change detection is analyzed using synthetic signal, logistic map and Lorenz map. The detection ability of CPE algorithm in different signal-to-noise ratios (SNR) is studied and the algorithm complexity is discussed. The results show that CPE can accurately capture minor feature information and amplify the detection results of dynamical changes compared with PE, weighted permutation entropy (WPE) and amplitude-aware permutation entropy (AAPE), but it has less robustness to noise and requires a higher computation cost than the others. Finally, we use the new algorithm to analyze the rolling bearing fault signals. The application of actual signals illustrates that CPE performs better in detecting abnormal pulse of the rolling bearing when the embedded dimension is small. From all the analyses in this paper, we find that CPE has a better performance for dynamical change detection compared with the other three algorithms when there is a larger repetition rate of permutation pattern in the position sequences.

Central limit theorems for patterns in multiset permutations and set partitions

We use the recently developed method of weighted dependency graphs to prove central limit theorems for the number of occurrences of any fixed pattern in multiset permutations and in set partitions. This generalizes results for patterns of size 2 in both settings, obtained by Canfield, Janson and Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.

Distributions of several infinite families of mesh patterns

Brändén and Claesson introduced mesh patterns to provide explicit expansions for certain permutation statistics as linear combinations of (classical) permutation patterns. The first systematic study of the avoidance of mesh patterns was conducted by Hilmarsson et al., while the first systematic study of the distribution of mesh patterns was conducted by the first two authors.In this paper, we provide far-reaching generalizations for 8 known distribution results and 5 known avoidance results related to mesh patterns by giving distribution or avoidance formulas for certain infinite families of mesh patterns in terms of distribution or avoidance formulas for smaller patterns. Moreover, as a corollary to a general result, we find the distribution of one more mesh pattern of length 2.

Pattern avoidance and quasisymmetric functions

Given a set of permutations Pi, let S_n(Pi) denote the set of permutations in the symmetric group S_n that avoid every element of Pi in the sense of pattern avoidance. Given a subset S of {1,...,n-1}, let F_S be the fundamental quasisymmetric function indexed by S. Our object of study is the generating function Q_n(Pi) = sum F_{Des sigma} where the sum is over all sigma in S_n(Pi) and Des sigma is the descent set of sigma. We characterize those Pi contained in S_3 such that Q_n(Pi) is symmetric or Schur nonnegative for all n. In the process, we show how each of the resulting Pi can be obtained from a theorem or conjecture involving more general sets of patterns. In particular, we prove results concerning symmetries, shuffles, and Knuth classes, as well as pointing out a relationship with the arc permutations of Elizalde and Roichman. Various conjectures and questions are mentioned throughout.

Separable elements in Weyl groups

We define the notion of a separable element in a finite Weyl group, generalizing the well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are rank-symmetric and rank-unimodal, and that the product of their rank generating functions gives that of the whole group, answering an open problem of Fan Wei. We also prove that separable elements are characterized by pattern avoidance in the sense of Billey and Postnikov.

Secure and Efficient Compressed Sensing-Based Encryption With Sparse Matrices

In this paper, we study the security of a compressed sensing (CS) based cryptosystem called a sparse one-time sensing (S-OTS) cryptosystem, which encrypts a plaintext with a sparse measurement matrix. To construct the secret matrix and renew it at each encryption, a bipolar keystream and a random permutation pattern are employed as cryptographic primitives, which can be obtained by a keystream generator of stream ciphers. With a small number of nonzero elements in the measurement matrix, the S-OTS cryptosystem achieves efficient CS encryption in terms of memory and computational cost. In security analysis, we show that the S-OTS cryptosystem can be indistinguishable as long as each plaintext has constant energy, which formalizes computational security against ciphertext only attacks (COA). In addition, we consider a chosen plaintext attack (CPA) against the S-OTS cryptosystem, which consists of two sequential stages, keystream and key recovery attacks. Against keystream recovery under CPA, we demonstrate that the S-OTS cryptosystem can be secure with overwhelmingly high probability, as an adversary needs to distinguish a prohibitively large number of candidate keystreams. Finally, we conduct an information-theoretic analysis to show that the S-OTS cryptosystem can be resistant against key recovery under CPA by guaranteeing that the probability of success is extremely low. In conclusion, the S-OTS cryptosystem can be computationally secure against COA and the two-stage CPA, while providing efficiency in CS encryption.

Patterns of relation triples in inversion and ascent sequences

Recently, Martinez and Savage carried out the systematic study of inversion sequences avoiding triples of relations. They reported many nice connections with familiar combinatorial families and posed several enumeration conjectures. All of their conjectures have already been solved except those related to the OEIS sequence A098746. In this paper, we address those remaining conjectures, which completes a picture for all the suspected connections arising in their investigation.As one of the most important subsets of inversion sequences, ascent sequences were introduced by Bousquet-Mélou et al. in bijection with (2+2)-free posets and their pattern avoidance properties have been extensively studied. We investigate some classical Eulerian and Stirling statistics on ascent sequences avoiding triples of relations. This leads us to find two new interpretations of the Catalan numbers and its refinements, and to interpret combinatorially a natural refinement of the binomial transformation of Catalan numbers. The latter discovery answers a challenging open problem posed by Pudwell.

On Pattern-Avoiding Fishburn Permutations

The class of permutations that avoid the bivincular pattern \((231, \{1\},\{1\})\) is known to be enumerated by the Fishburn numbers. In this paper, we call them Fishburn permutations and study their pattern avoidance. For classical patterns of size 3, we give a complete enumerative picture for regular and indecomposable Fishburn permutations. For patterns of size 4, we focus on a Wilf equivalence class of Fishburn permutations that are enumerated by the Catalan numbers. In addition, we also discuss a class enumerated by the binomial transform of the Catalan numbers and give conjectures for other equivalence classes of pattern-avoiding Fishburn permutations.

Permutation patterns in genome rearrangement problems: The reversal model

In the context of the genome rearrangement problem, we analyze two well known models, namely the reversal and the prefix reversal models, by exploiting the connection with the notion of permutation patterns. More specifically, for any k, we provide a characterization of the set of permutations having distance less than or equal to k from the identity (which is known to be a permutation class) in terms of what we call generating peg permutations and we describe some properties of its basis, which allow us to compute such a basis for small values of k.

Enumeration of permutations avoiding a triple of 4-letter patterns is almost all done

Abstract This paper basically completes a project to enumerate permutations avoiding a triple T of 4-letter patterns, in the sense of classical pattern avoidance, for every T. There are 317 symmetry classes of such triples T and previous papers have enumerated avoiders for all but 14 of them. One of these 14 is conjectured not to have an algebraic generating function. Here, we find the generating function for each of the remaining 13, and it is algebraic in each case.

Resolving Two Conjectures on Staircase Encodings and Boundary Grids of $132$ and $123$-avoiding permutations

This paper analyzes relations between pattern avoidance of certain permutations and graphs on staircase grids and boundary grids, and proves two conjectures posed by Bean, Tannock, and Ulfarsson (2015). More specifically, this paper enumerates a certain family of staircase encodings and proves that the downcore graph, a certain graph established on the boundary grid, is pure if and only if the permutation corresponding to the boundary grid avoids the classical patterns 123 and 2143.

Supercritical sequences, and the nonrationality of most principal permutation classes

We prove that for any fixed n, and for most permutation patterns q, the number Avn,ℓ(q) of q-avoiding permutations of length n that consist of ℓ skew blocks is a monotone decreasing function of ℓ. We then show that this implies that for most patterns q, the generating function ∑n≥0Avn(q)zn of the sequence Avn(q) of the numbers of q-avoiding permutations is not rational. Placing our results in a broader context, we show that for rational power series F(z) and G(z) with nonnegative real coefficients, the relation F(z)=1∕(1−G(z)) is supercritical, while for most permutation patterns q, the corresponding relation is not supercritical.

Permutation Entropy and Irreversibility in Gait Kinematic Time Series from Patients with Mild Cognitive Decline and Early Alzheimer’s Dementia

Gait is a basic cognitive purposeful action that has been shown to be altered in late stages of neurodegenerative dementias. Nevertheless, alterations are less clear in mild forms of dementia, and the potential use of gait analysis as a biomarker of initial cognitive decline has hitherto mostly been neglected. Herein, we report the results of a study of gait kinematic time series for two groups of patients (mild cognitive impairment and mild Alzheimer’s disease) and a group of matched control subjects. Two metrics based on permutation patterns are considered, respectively measuring the complexity and irreversibility of the time series. Results indicate that kinematic disorganisation is present in early phases of cognitive impairment; in addition, they depict a rich scenario, in which some joint movements display an increased complexity and irreversibility, while others a marked decrease. Beyond their potential use as biomarkers, complexity and irreversibility metrics can open a new door to the understanding of the role of the nervous system in gait, as well as its adaptation and compensatory mechanisms.

Vincular patterns in inversion sequences

Inversion sequences have intriguing applications in combinatorics, computer sciences and polyhedral geometry. Ascent sequences, as one of the most important subsets of inversion sequences, were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev to encode the (2+2)-free posets. Pattern avoidance in ascent sequences was first studied by Ducan and Steingrímsson in 2011, while the systematic study of patterns in inversion sequences was initiated only recently by Corteel–Martinez–Savage–Weselcouch and Mansour–Shattuck. In this paper, we investigate systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length 3, where two of the three letters are required to be adjacent. Our results connect restricted inversion sequences and ascent sequences to a number of well-known combinatorial sequences including Bell numbers, Fishburn numbers, Powered Catalan numbers, Semi-Baxter numbers and the number of 3-noncrossing partitions. A variety of combinatorial bijections, known or newly constructed, are applied to achieve these interesting connections.

On Partially Ordered Patterns of Length 4 and 5 in Permutations

Partially ordered patterns (POPs) generalize the notion of classical patterns studied widely in the literature in the context of permutations, words, compositions and partitions. In an occurrence of a POP, the relative order of some of the elements is not important. Thus, any POP of length $k$ is defined by a partially ordered set on $k$ elements, and classical patterns correspond to $k$-element chains. The notion of a POP provides a convenient language to deal with larger sets of permutation patterns.
 This paper contributes to a long line of research on classical permutation patterns of length 4 and 5, and beyond, by conducting a systematic search of connections between sequences in the Online Encyclopedia of Integer Sequences (OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i) obtain 13 new enumerative results for classical patterns of length 4 and 5, and a number of results for patterns of arbitrary length, (ii) collect under one roof many sporadic results in the literature related to avoidance of patterns of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the most intriguing bijective questions we state, 7 are related to explaining Wilf-equivalence of various sets of patterns, e.g. 5 or 8 patterns of length 4, and 2 or 6 patterns of length 5.

Occurrence graphs of patterns in permutations

We define the \emph{occurrence graph} $G_p(\pi$) of a pattern $p$ in a permutation $\pi$ as the graph with the occurrences of $p$ in $\pi$ as vertices and edges between the vertices if the occurrences differ by exactly one element. We then study properties of these graphs. The main theorem in this paper is that every \emph{hereditary property} of graphs gives rise to a \emph{permutation class}.

Asymptotics of pattern avoidance in the Klazar set partition and permutation-tuple settings

We consider asymptotics of set partition pattern avoidance in the sense of Klazar. Our main result derives the asymptotics of the number of set partitions avoiding a given set partition within an exponential factor, which leads to a classification of possible growth rates of set partition pattern classes. We further define a notion of permutation-tuple avoidance, which generalizes notions of Aldred et al. and the usual permutation pattern setting, and similarly determine the number of permutation-tuples avoiding a given tuple to within an exponential factor. We note a seeming connection to previous results on hereditary properties of labelled graphs, prompting the question of if there is a generalization to ordered graphs.

Pattern avoidance in permutations and their squares

We study permutations p such that both p and p2 avoid a given pattern q. We obtain a generating function for the case of q=312 (equivalently, q=231), we prove that if q is monotone increasing, then above a certain length, there are no such permutations, and we prove an upper bound for q=321. We also present some intriguing questions in the case of q=132.

Mahonian STAT on rearrangement class of words

In 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discovered by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on R(w), the rearrangement class of a given word w. We construct such an involution in this paper, which yields a comparable joint equidistribution between two sextuple statistics over R(w). Our involution builds on Burstein’s involution and Foata–Schützenberger’s involution that utilizes the celebrated RSK algorithm.

Patterns in Random Permutations Avoiding Some Sets of Multiple Patterns

We consider a random permutation drawn from the set of permutations of length n that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern sigma has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers.