Class Of Permutations Research Articles

An Analysis of the 𝒞 Class of Bent Functions

Two (so-called C;D) classes of permutation-based bent Boolean functions were introduced by Carlet two decades ago, but without specifying some explicit construction methods for their construction (apart from the subclass D0). In this article, we look in more detail at theC class, and derive some existence and nonexistence results concerning the bent functions in the C class for many of the known classes of permutations over F2n. Most importantly, the existence results induce generic methods of constructing bent functions in classC which possibly do not belong to the completed Maiorana-McFarland class. The question whether the specic permutations and related subspaces we identify in this article indeed give bent functions outside the completed Maiorana-McFarland class remains open.

A novel sensitive image encryption algorithm based on the Zaslavsky chaotic map

ABSTRACTIn this article, a novel sensitive encryption scheme to secure the digital images based on the Zaslavsky chaotic map is proposed. We employ the Zaslavsky chaotic map as a pseudo-random generator to produce the key encryption of the proposed image cryptosystem. The cipher structure has been chosen based on permutation-diffusion processes, where we adopt the classic permutation substitution network, which ensures both confusion and diffusion properties for the encrypted image. Our proposed algorithm has high sensitivity in plain image and the secret key. Moreover, the results show that the characteristics of our approach have excellent performance, with high scores (NPRC = 99.61%, UACI = 33.47%, entropy (CipherImage) 8, and correlation coefficient 0). Experimental results have been studied and analyzed in detail with various types of security analysis. These results demonstrate that our proposed cryptosystem has highly satisfactory security performance and can withstand various attacks compared to state-of-the-art methods.

Infinite classes of vectorial plateaued functions, permutations and complete permutations

We use the well-known Maiorana–McFarland class to construct several important combinatorial structures. In the first place, we easily identify infinite classes of vectorial plateaued functions {F}:F2n→F2n such that all non-zero linear combinations of its component functions are also plateaued. More importantly, by setting certain restrictions on the component functions, the same approach also yields many infinite classes of permutations for any n≥6. Finally, we deduce some infinite classes of complete permutations, as a subclass of these permutations. Most notably, all these classes are of variable and controllable degree, the property being intrinsic to the construction method. The construction method is highly tweakable giving rise to many variations that again provide us with infinite classes of these structures.

Major index distribution over permutation classes

For a permutation π the major index of π is the sum of all indices i such that πi>πi+1. It is well known that the major index is equidistributed with the number of inversions over all permutations of length n. In this paper, we study the distribution of the major index over pattern-avoiding permutations of length n. We focus on the number Mnm(Π) of permutations of length n with major index m, avoiding the set of patterns Π.First we are able to show that for a singleton set Π={σ} other than some trivial cases, the values Mnm(Π) are monotonic in the sense that Mnm(Π)≤Mn+1m(Π). Our main result is a study of the asymptotic behaviour of Mnm(Π) as n goes to infinity. We prove that for every fixed m, Π and n large enough, Mnm(Π) is equal to a polynomial in n and moreover, we are able to determine the degrees of these polynomials for many sets of patterns.

Unsplittable Classes of Separable Permutations

A permutation class is splittable if it is contained in the merge of two of its proper subclasses. We characterise the unsplittable subclasses of the class of separable permutations both structurally and in terms of their bases.

Asymptotics for minimal overlapping patterns for generalized Euler permutations, standard tableaux of rectangular shape, and column strict arrays

A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal overlapping patterns in $S_j$ is at least $3 -e$. Given a permutation $\sigma$, we let $\text{Des}(\sigma)$ denote the set of descents of $\sigma$. We study the class of permutations $\sigma \in S_{kn}$ whose descent set is contained in the set $\{k,2k, \ldots (n-1)k\}$. For example, up-down permutations in $S_{2n}$ are the set of permutations whose descent equal $\sigma$ such that $\text{Des}(\sigma) = \{2,4, \ldots, 2n-2\}$. There are natural analogues of the minimal overlapping permutations for such classes of permutations and we study the proportion of minimal overlapping patterns for each such class. We show that the proportion of minimal overlapping permutations in such classes approaches $1$ as $k$ goes to infinity. We also study the proportion of minimal overlapping patterns in standard Young tableaux of shape $(n^k)$.

Avoiding vincular patterns on alternating words

A word w=w1w2⋯wn is alternating if either w1<w2>w3<w4>⋯ (when the word is up-down) or w1>w2<w3>w4<⋯ (when the word is down-up). The study of alternating words avoiding classical permutation patterns was initiated by the authors in Gao et al. [6], where, in particular, it was shown that 123-avoiding up-down words of even length are counted by the Narayana numbers.However, not much was understood on the structure of 123-avoiding up-down words. In this paper, we fill in this gap by introducing the notion of a cut-pair that allows us to subdivide the set of words in question into equivalence classes. We provide a combinatorial argument to show that the number of equivalence classes is given by the Catalan numbers, which induces an alternative (combinatorial) proof of the corresponding result in Gao et al. [6].Further, we extend the enumerative results in Gao et al. [6] to the case of alternating words avoiding a vincular pattern of length 3. We show that it is sufficient to enumerate up-down words of even length avoiding the consecutive pattern 132¯ and up-down words of odd length avoiding the consecutive pattern 312¯ to answer all of our enumerative questions. The former of the two key cases is enumerated by the Stirling numbers of the second kind.

Locally Convex Words and Permutations

We introduce some new classes of words and permutations characterized by the second difference condition $\pi(i-1) + \pi(i+1) - 2\pi(i) \leq k$, which we call the $k$-convexity condition. We demonstrate that for any sized alphabet and convexity parameter $k$, we may find a generating function which counts $k$-convex words of length $n$. We also determine a formula for the number of 0-convex words on any fixed-size alphabet for sufficiently large $n$ by exhibiting a connection to integer partitions. For permutations, we give an explicit solution in the case $k = 0$ and show that the number of 1-convex and 2-convex permutations of length $n$ are $\Theta(C_1^n)$ and $\Theta(C_2^n)$, respectively, and use the transfer matrix method to give tight bounds on the constants $C_1$ and $C_2$. We also providing generating functions similar to the the continued fraction generating functions studied by Odlyzko and Wilf in the "coins in a fountain" problem.

Linear Clique‐Width for Hereditary Classes of Cographs

AbstractThe class of cographs is known to have unbounded linear clique‐width. We prove that a hereditary class of cographs has bounded linear clique‐width if and only if it does not contain all quasi‐threshold graphs or their complements. The proof borrows ideas from the enumeration of permutation classes.

An Erd\H{o}s--Hajnal analogue for permutation classes

Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$.

Counting and Generating Permutations in Regular Classes

The signature of a permutation $$\sigma $$ź is a word $$\mathtt {sg}(\sigma )\subseteq \{\mathfrak {a},\mathfrak {d}\}^*$$sg(ź)⊆{a,d}ź with ith letter $$\mathfrak {d}$$d when $$\sigma $$ź has a descent [i.e. $$\sigma (i)>\sigma (i+1)$$ź(i)>ź(i+1)] and $$\mathfrak {a}$$a when $$\sigma $$ź has an ascent [i.e. $$\sigma (i)<\sigma (i+1)$$ź(i)<ź(i+1)]. The combinatorics of permutations with a prescribed signature is quite well explored. Here we introduce regular classes of permutations, the sets $$\varLambda (L)$$ź(L) of permutations with signature in regular languages $$L\subseteq \{\mathfrak {a},\mathfrak {d}\}^*$$L⊆{a,d}ź. Given a regular class of permutations we (i) count the permutations of a given length within the class; (ii) compute a closed form formula for the exponential generating function; and (iii) sample uniformly at random the permutation of a given length. We first recall how (i) is solved in the literature for the case of a single signature. We then explain how to extend these methods to regular classes of permutations using language equations from automata theory. We give two methods to solve (ii) in terms of exponentials of matrices. For the third problem we provide both discrete and continuous recursive methods as well as an extension of Boltzmann sampling to uncountable union of sets parametrised by a variable ranging over an interval. Last but not least, a part of our contributions are based on a geometric interpretation of a subclass of regular timed languages (that is, recognised by timed automata specific to our problem).

The Brownian limit of separable permutations

We study uniform random permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a “Brownian separable permuton”.

A new family of differentially 4-uniform permutations over $$\mathbb{F}_{2^{2k} }$$ for odd k

We study the differential uniformity of a class of permutations over $\mathbb{F}_{2^{n}}$ with $n$ even. These permutations are different from the inverse function as the values $x^{-1}$ are modified to be $(\gamma x)^{-1}$ on some cosets of a fixed subgroup $\langle\gamma\rangle$ of $\mathbb{F}_{2^{n}}^*$. We obtain some sufficient conditions for this kind of permutations to be differentially 4-uniform, which enable us to construct a new family of differentially 4-uniform permutations that contains many new Carlet-Charpin-Zinoviev equivalent (CCZ-equivalent) classes as checked by Magma for small numbers $n$. Moreover, all of the newly constructed functions are proved to possess optimal algebraic degree and relatively high nonlinearity.

Tauberian Theorem for Generating Functions of Multiple Series

Multiple sequences $a(i)\ge0$, $i\in {\bf Z}_+^n$, are considered. We introduce the concept of one-sided weak oscillation of such sequences along the sequence $(m=m(k)=(m_1(k),\dots,m_n(k)))$, $m_j(k)>0$ $\forall j=1,\dots,n$, $k\in {\bf N}$, such that $m_j(k)\to\infty$ as $k\to\infty$. From asymptotics of the generating function $A(s)$, $s\in[0,1)^n$, of a studied multiple sequence for $s=(e^{-\lambda_1/m_1},\dots,e^{-\lambda_n/m_n})$ as $k\to\infty$ ($\lambda_1,\dots,\lambda_n$ are positive and fixed), we deduce the asymptotics of $a(x_1m_1,\dots,x_nm_n)$ (the numbers $x_1,\dots,x_n$ are also positive and fixed). The Tauberin theorem obtained here generalizes some Tauberian theorems previously proved by the author earlier in research of some classes of random permutations and random mappings of a finite set into it. Thus, the initial result in this direction is the well-known Jovan Karamata's Tauberian theorem for the generating functions of the sequences.

Graphs emerging from the solutions to the periodic discrete Toda equation over finite fields

The periodic discrete Toda equation defined over finite fields has been studied. We obtained the finite graph structures constructed by the network of states where edges denote possible time evolutions. We simplify the graphs by introducing a equivalence class of cyclic permutations to the initial values. We proved that the graphs are bi-directional and that they are composed of several arrays of complete graphs connected at one of their vertices.

A Construction of Permutation Codes From Rational Function Fields and Improvement to the Gilbert–Varshamov Bound

Due to recent applications to communications over powerlines, multilevel flash memories, and block ciphers, permutation codes have received a lot of attention from both coding and mathematical communities. One of the benchmarks for good permutation codes is the Gilbert–Varshamov bound. Although there have been several constructions of permutation codes, the Gilbert–Varshamov bound still remains to be the best asymptotical lower bound except for a recent improvement in the case of constant minimum distance. In this paper, we present an algebraic construction of permutation codes from rational function fields, and it turns out that, for a prime number $n$ of a symbol length, this class of permutation codes improves the Gilbert–Varshamov bound by a factor $n$ asymptotically for a minimum distance $d$ with $d=O(\sqrt {n})$ . Furthermore, for a constant minimum distance $d$ , we improve the Gilbert–Varshamov bound by a factor $n$ as well as the recent one given by Gao et al. by a factor $n/\log n$ asymptotically for all sufficiently large $n$ .

On the effective and automatic enumeration of polynomial permutation classes

We describe an algorithm, implemented in Python, which can enumerate any permutation class with polynomial enumeration from a structural description of the class. In particular, this allows us to find formulas for the number of permutations of length n which can be obtained by a finite number of block sorting operations (e.g., reversals, block transpositions, cut-and-paste moves).

A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes

Rauzy Classes and Extended Rauzy Classes are equivalence classes of permutations that arise when studying Interval Exchange Transformations.In 2003, Kontsevich and Zorich classified Extended Rauzy Classes by using data from Translation Surfaces, which are associated to IET's thanks to the Zippered Rectangle Construction of Veech from 1982.In 2009, Boissy finalized the classification of Rauzy Classes also using information from Translation Surfaces.We present in this paper specialized moves in (Extended) Rauzy Classes that allow us to prove the sufficiency and necessity in the previous classification theorems.These results provide a complete, and purely combinatorial, proof of these known results.We end with some general statements about our constructed move.

Single machine interfering jobs problem with flowtime objective

Interfering jobs problems or multi agents scheduling problems are an emergent topic in the scheduling literature. In this problem two or more sets of jobs are scheduled, each one with its own criterion. We study here the problem occurring when jobs from two sets are scheduled on a single machine. The objective is to minimize the total flowtime of one set, while maintaining the total flowtime of the jobs in the other set lower or equal than a given constant. This problem is known to be NP-hard. We provide some properties and code the solutions according to these properties using a binary codification. We compare this binary codification with the classical permutation codification using a Genetic Algorithm. We have tested small and big instances, and show the advantages up the binary codification.

Permutation classes and polyomino classes with excluded submatrices

This article introduces an analogue of permutation classes in the context of polyominoes. For both permutation classes and polyomino classes, we present an original way of characterizing them by avoidance constraints (namely, with excluded submatrices) and we discuss how canonical such a description by submatrix-avoidance can be. We provide numerous examples of permutation and polyomino classes which may be defined and studied from the submatrix-avoidance point of view, and conclude with various directions for future research on this topic.