Class Of Permutations Research Articles

Joint distributions of statistics over permutations avoiding two patterns of length 3

Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al. found distributions of ascents and descents on permutations avoiding any two patterns of length 3. In this paper, we generalize these results in two different ways: we find explicit formulas for the joint distribution of six statistics (asc, des, lrmax, lrmin, rlmax, rlmin), and also explicit formulas for the joint distribution of four statistics (asc, des, MNA, MND) on these permutations in all cases. The latter result also extends the recent studies by Kitaev and Zhang of the statistics MNA and MND (related to non-overlapping occurrences of ascents and descents) on stack-sortable permutations. All multivariate generating functions in our paper are rational, and we provide combinatorial proofs of five equidistribution results that can be derived from the generating functions.

Sorting genomes by prefix double-cut-and-joins

In this paper, we study the problem of sorting unichromosomal linear genomes by prefix double-cut-and-joins (or DCJs) in both the signed and the unsigned settings. Prefix DCJs cut the leftmost segment of a genome and any other segment, and recombine the severed endpoints in one of two possible ways: one of these options corresponds to a prefix reversal, which reverses the order of elements between the two cuts (as well as their signs in the signed case). Our main results are: (1) new structural lower bounds based on the breakpoint graph for sorting by unsigned prefix reversals, unsigned prefix DCJs, and signed prefix DCJs; (2) two polynomial-time algorithms for sorting by prefix DCJs, both in the signed case (which answers an open question of Labarre [1]) and in the unsigned case; (3) a 1-absolute approximation algorithm for sorting by unsigned prefix reversals for a specific class of permutations.

Mini-Workshop: Permutation Patterns

The study of permutation patterns has recently seen several surprising results, and the purpose of this mini-workshop was to bring together researchers from across the field to focus on four hot topics related to these recent developments. The topics covered the nature of generating functions that enumerate permutation classes, the structure of permutation classes and the impact this has on their growth rates, and the study of permutons, which lies at the interface of permutation patterns and discrete probability. The workshop offered an opportunity for knowledge exchange, but also time and space to initiate group collaborations on open problems related to these topics.

Arithmetization-oriented APN permutations

AbstractRecently, many cryptographic primitives such as homomorphic encryption (HE), multi-party computation (MPC) and zero-knowledge (ZK) protocols have been proposed in the literature which operate on the prime field $${\mathbb {F}}_p$$ F p for some large prime p. Primitives that are designed using such operations are called arithmetization-oriented primitives. As the concept of arithmetization-oriented primitives is new, a rigorous cryptanalysis of such primitives is yet to be done. In this paper, we investigate arithmetization-oriented APN functions. More precisely, we investigate APN permutations in the CCZ-classes of known families of APN power functions over the prime field $${\mathbb {F}}_p$$ F p . Moreover, we present a class of binomial permutation having differential uniformity at most 5 defined via the quadratic character over finite fields of odd characteristic. Computationally it is confirmed that the latter family contains new APN permutations for some small parameters. We conjecture it to contain an infinite subfamily of APN permutations.

Long increasing subsequences and non-algebraicity

We use a recent result of Alin Bostan to prove that the generating functions of two infinite sequences of permutation classes are not algebraic.

Circular code identified by the codon usage

Since 1996, circular codes in genes have been identified thanks to the development of 6 statistical approaches: trinucleotide frequencies per frame (Arquès and Michel, 1996), correlation functions per frame (Arquès and Michel, 1997), frame permuted trinucleotide frequencies (Frey and Michel, 2003, 2006), advanced statistical functions at the gene population level (Michel, 2015) and at the gene level (Michel, 2017). All these 3-frame statistical methods analyse the trinucleotide information in the 3 frames of genes: the reading frame and the 2 shifted frames. Notably, codon usage does not allow for the identification of circular codes (Michel, 2020). This has been a long-standing problem since 1996, hindering biologists’ access to circular code theory.By considering circular code conditions resulting from code theory, particularly the concept of permutation class, and building upon previous statistical work, a new statistical approach based solely on the codon usage, i.e. a 1-frame statistical method, surprisingly reveals the maximal C3 self-complementary trinucleotide circular code X in bacterial genes and in average (bacterial, archaeal, eukaryotic) genes, and almost in archaeal genes. Additionally, a new parameter definition indicates that bacterial and archaeal genes exhibit codon usage dispersion of the same order of magnitude, but significantly higher than that observed in eukaryotic genes. This statistical finding may explain the greater variability of codes in eukaryotic genes compared to bacterial and archaeal genes, an issue that has been open for many years. Finally, biologists can now search for new (variant) circular codes at both the genome level (across all genes in a given genome) and the gene level using only codon usage, without the need for analysing the shifted frames.

Twin-width and permutations

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been further extended to binary structures, in several (basically equivalent) ways. We prove that a class of binary relational structures (that is: edge-colored partially directed graphs) has bounded twin-width if and only if it is a first-order transduction of a~proper permutation class. As a by-product, we show that every class with bounded twin-width contains at most $2^{O(n)}$ pairwise non-isomorphic $n$-vertex graphs.

Half-exceeded Symmetric Permutations and the Inverse Pairs

In this paper, we introduce a class of restricted symmetric permutations, called half-exceeded symmetric permutations. We deduce the enumerative formula of the permutations of { 1 , 2 , … , 2 n } and give it a refinement according to the distribution of the inverse pairs. As a consequence, we obtain new combinatorial interpretations of some well-known sequences, such as Stirling numbers of the second kind and ordered Bell numbers. Moreover, we introduce the ordered Stirling number of the second kind and establish a combinatorial proof of the recursive relation of the sequence.

Sorting via shuffles with a cut after the longest increasing prefix

We define four new shuffling algorithms on permutations. For each algorithm, we characterize and enumerate the sets of permutations that are sorted after k iterations of the algorithm and we determine properties of the corresponding generating functions. For three of the algorithms, the sets of sortable permutations can be seen to be permutation classes for any nonnegative integer k.

Labelled Well-Quasi-Order in Juxtapositions of Permutation Classes

Labelled Well-Quasi-Order in Juxtapositions of Permutation Classes

Positroid Catalan numbers

Given a (bounded affine) permutation f f , we study the positroid Catalan number C f C_f defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated q , t q,t -polynomials coincide with the generalized q , t q,t -Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov–Rozansky homology of Coxeter links.

A logical limit law for $231$-avoiding permutations

We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $\Psi$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,\Psi}$: (i) it is either bounded away from $0$, or decays exponentially fast; (ii) the set of possible limits is dense in $[0,1]$. Our tools come mainly from analytic combinatorics and singularity analysis.

Sorting with a popqueue

We introduce a new sorting device for permutations, which we call popqueue. It consists of a special queue, having the property that any time one wants to extract elements from the queue, actually all the elements currently in the queue are poured into the output. We illustrate two distinct optimal algorithms, called Min and Cons, to sort a permutation using such a device, which allow us also to characterize sortable permutations in terms of pattern avoidance. We next investigate what happens by making two passes through a popqueue, showing that the set of sortable permutations is not a class for Min, whereas it is for Cons. In the latter case we also explicitly find the basis of the class of sortable permutations. Finally, we study preimages under Cons (by means of an equivalent version of the algorithm), and find a characterization of the set of preimages of a given permutation. We also give some enumerative results concerning the number of permutations having k preimages, for k = 1, 2, 3, and we conclude by observing that there exist permutations having k preimages for any value of k ≥ 0.

On the Use of a Complex Indicator of the Stability of Permutation Entropy of Time Series Fragments When Analyzing Infrasound Monitoring Signals of the Altai Republic

This paper discusses one of the approaches that allows us to assess the degree of complexity or randomness of fragments of a time series in order to detect infrasound or geomagnetic signals in the results of observations of the dynamics of the natural or man-made processes under study. In our case, we are talking about monitoring the infrasound background on the territory of the Altai Republic. To solve the problem of estimating the required characteristics of a time series with minimal computational costs and in real time, a complex indicator of the stability of permutation entropy is introduced, since estimating the value of classical permutation entropy for n = 3 (the most commonly used version of permutation entropy) does not allow solving the problem with sufficient accuracy.

Calibrated weighted permutation test detects ancient language connections in the Circumpolar area (Chukotian-Nivkh and Yukaghir-Samoyedic)*

Abstract Relationships between universally recognized language families represent a hotly debated topic in historical linguistics, and the same is true for correlation between signals of genetic and linguistic relatedness. We developed a weighted permutation test which represents the classical permutation tests with weights introduced for individual Swadesh concepts according to their typological stability. Further, the obtained values were calibrated on a negative control group to override non-uniform distribution of phonemes within the Swadesh wordlist. We applied the calibrated permutation test to the basic vocabularies of nine languages and reconstructed proto-languages to show that three groups of circumpolar language families in the Northern Hemisphere show evidence of relationship through common descent or borrowing in the basic vocabulary: [Chukotko-Kamchatkan, Nivkh]; [Yukaghir, Samoyedic]; and [Yeniseian, Na-Dene, Burushaski]. The former two pairs showed the most significant signals of language relationship. Our findings further support some hypotheses on long-distance language relationships previously put forward based on linguistic methods but lacking universal acceptance.

A lightweight symmetric image encryption cryptosystem in wavelet domain based on an improved sine map

In the era of big data, the number of images transmitted over the public channel increases exponentially. As a result, it is crucial to devise the efficient and highly secure encryption method to safeguard the sensitive image. In this paper, an improved sine map (ISM) possessing a larger chaotic region, more complex chaotic behavior and greater unpredictability is proposed and extensively tested. Drawing upon the strengths of ISM, we introduce a lightweight symmetric image encryption cryptosystem in wavelet domain (WDLIC). The WDLIC employs selective encryption to strike a satisfactory balance between security and speed. Initially, only the low-frequency-low-frequency component is chosen to encrypt utilizing classic permutation and diffusion. Then leveraging the statistical properties in wavelet domain, Gaussianization operation which opens the minds of encrypting image information in wavelet domain is first proposed and employed to all sub-bands. Simulations and theoretical analysis demonstrate the high speed and the remarkable effectiveness of WDLIC.

The Complete Differential Spectrum of a Class of Power Permutations Over Odd Characteristic Finite Fields

The Complete Differential Spectrum of a Class of Power Permutations Over Odd Characteristic Finite Fields

A Bijection Between Evil-Avoiding and Rectangular Permutations

Evil-avoiding permutations, introduced by Kim and Williams in 2022, arise in the study of the inhomogeneous totally asymmetric simple exclusion process. Rectangular permutations, introduced by Chirivì, Fang, and Fourier in 2021, arise in the study of Schubert varieties and Demazure modules. Taking a suggestion of Kim and Williams, we supply an explicit bijection between evil-avoiding and rectangular permutations in $S_n$ that preserves the number of recoils. We encode these classes of permutations as regular languages and construct a length-preserving bijection between words in these regular languages. We extend the bijection to another Wilf-equivalent class of permutations, namely the $1$-almost-increasing permutations, and exhibit a bijection between rectangular permutations and walks of length $2n-2$ in a path of seven vertices starting and ending at the middle vertex.

Distributional Chaos and Sensitivity for a Class of Cyclic Permutation Maps

Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p-dimensional dynamical systems of the form φ(b1,b2,⋯,bp)=(up(bp),u1(b1),⋯,up−1(bp−1)), where bj∈Hj (j∈{1,2,⋯,p}), p≥2 is an integer, and Hj (j∈{1,2,⋯,p}) are compact subintervals of the real line R=(−∞,+∞). uj:Hj→Hj+1(j=1,2,…,p−1) and up:Hp→H1 are continuous maps. Necessary and sufficient conditions for a class of cyclic permutation maps to have Li–Yorke chaos, distributional chaos in a sequence, distributional chaos, or Li–Yorke sensitivity are given. These results extend the existing ones.

The Number of Occurrences of Patterns in a Random Tree or Forest Permutation

The classes of tree permutations and forest permutations were defined by Acan and Hitczenko (2016). We study random permutations of a given length from these classes, and in particular the number of occurrences of a fixed pattern in one of these random permutations. The main results show that the distributions of these numbers are asymptotically normal.
 The proof uses representations of random tree and forest permutations that enable us to express the number of occurrences of a pattern by a type of U-statistics; we then use general limit theorems for the latter.