Class Of Permutations Research Articles

On the constructions of n-cycle permutations

Any permutation polynomial is an n-cycle permutation. When n is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for n-cycle permutations, which mainly are of the form xrh(xs). We then propose unified constructing methods including recursive ways and a cyclotomic way for n-cycle permutations of such form. We demonstrate our approaches by constructing three classes of explicit triple-cycle permutations with high index and two classes of n-cycle permutations with low index, many of which are new both at levels of permutation property and cycle property.

Differential spectra of a class of power permutations with characteristic 5

Let n be a positive integer and $$F(x)=x^d$$ over $${\mathbb {F}}_{5^n}$$ , where $$d=\frac{5^n-3}{2}$$ . In this paper, we study the differential properties of the power permutation F(x). It is shown that F(x) is differentially 4-uniform when n is even, and differentially 5-uniform when n is odd. Based on some knowledge on elliptic curves over finite fields, the differential spectrum of F(x) is also determined.

Enumeration of Permutation Classes and Weighted Labelled Independent Sets

In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $\mathrm{Av}(2314,3124)$, $\mathrm{Av}(2413,3142)$, $\mathrm{Av}(2413,3124)$, $\mathrm{Av}(2413,2134)$ and $\mathrm{Av}(2314,2143)$, as well as many subclasses.

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

AbstractWe consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one‐dimensional‐labeled generating tree (together with some technical assumptions) and implies local convergence for random permutations in such families. We exhibit ten different families of permutations, most of them being permutation classes, that satisfy our assumptions. To the best of our knowledge, this is the first work where generating trees—which were introduced to enumerate combinatorial objects—have been used to establish probabilistic results.

Stack-sorting with consecutive-pattern-avoiding stacks

We introduce consecutive-pattern-avoiding stack-sorting maps SCσ, which are natural generalizations of West's stack-sorting map s and natural analogues of the classical-pattern-avoiding stack-sorting maps sσ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns σ such that Sort(SCσ), the set of permutations that are sortable via the map s∘SCσ, is a permutation class, and we enumerate the sets Sort(SCσ) for σ∈{123,132,321}. We also study the maps SCσ from a dynamical point of view, characterizing the periodic points of SCσ for all σ∈S3 and computing maxπ∈Sn⁡|SCσ−1(π)| for all σ∈{132,213,231,312}. In addition, we characterize the periodic points of the classical-pattern-avoiding stack-sorting map s132, and we show that the maximum number of iterations of s132 needed to send a permutation in Sn to a periodic point is n−1. The paper ends with numerous open problems and conjectures.

An image encryption scheme based on double chaotic cyclic shift and Josephus problem

In recent years, image encryption has become a very popular and effective method to protect digital images. Among many image encryption methods, the image encryption algorithm based on chaotic map has extensive application and profound theoretical basis. In this paper, a novel image encryption algorithm is proposed by combining chaotic map with Josephus problem. The whole encryption process adopts the classical permutation–diffusion structure. In the permutation process, the double chaotic cycle shift algorithm has been proposed by improving the chaotic shift transform method. In the diffusion stage, we extend the definition of Josephus problem by using chaotic map to preserve the Josephus sequence diversity. We then divide the image into sub-blocks and pick one randomly according to Josephus sequence. Furthermore, we change the pixel values of the image through performing modular addition and XOR operations on the pixels between blocks. Through the above process, we get the image encryption algorithm based on double chaotic cycle shift and Josephus problem (JP-DCCS). In the simulation experiment and security analysis, we compare JP-DCCS with the other advanced image encryption algorithms. The results show that JP-DCCS has better performance and higher security.

Graph universal cycles of combinatorial objects

A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as ucycles or generalized deBruijn cycles or U-cycles) of several combinatorial objects. The existence of ucycles is often dependent on the specific representation that we use for the combinatorial objects. For example, should we represent the subset {2,5} of {1,2,3,4,5} as “25” in a linear string? Is the representation “52” acceptable? Or is it tactically advantageous (and acceptable) to go with {0,1,0,0,1}? In this paper, we represent combinatorial objects as graphs, as in [3], and exhibit the flexibility and power of this representation to produce graph universal cycles, or Gucycles, for k-subsets of an n-set; permutations (and classes of permutations) of [n]={1,2,…,n}, and partitions of an n-set, thus revisiting the classes first studied in [5]. Under this graphical scheme, we will represent {2,5} as the subgraph A of C5 with edge set consisting of {2,3} and {5,1}, namely the “second” and “fifth” edges in C5. Permutations are represented via their permutation graphs, and set partitions through disjoint unions of complete graphs.

The relation between composability and splittability of permutation classes

A permutation class C is said to be splittable if there exist two proper subclasses A,B⊊C such that any σ∈C can be red–blue colored so that the red (respectively, blue) subsequence of σ is order isomorphic to an element of A (respectively, B). The class C is said to be composable if there exists some number of proper subclasses A1,…,Ak⊊C such that any σ∈C can be written as α1∘⋯∘αk for some αi∈Ai. We answer a question of Karpilovskij by showing that there exists a composable permutation class that is not splittable. We also give a condition under which an infinite composable class must be splittable.

R-Local Unlabeled Sensing: A Novel Graph Matching Approach for Multiview Unlabeled Sensing Under Local Permutations

Unlabeled sensing is a linear inverse problem where the measurements are scrambled under an unknown permutation leading to loss of correspondence between the measurements and the rows of the sensing matrix. Motivated by practical tasks such as mobile sensor networks, target tracking and the pose and correspondence estimation between point clouds, we study a special case of this problem restricting the class of permutations to be local and allowing for multiple views. In this setting, namely unlabeled multi-view sensing with local permutation, previous results and algorithms are not directly applicable. In this paper, we propose a computationally efficient algorithm that creatively exploits the machinery of graph alignment and Gromov-Wasserstein alignment and leverages the multiple views to estimate the local permutations. Simulation results on synthetic data sets indicate that the proposed algorithm is scalable and applicable to the challenging regimes of low to moderate SNR.

Separable elements and splittings of Weyl groups

We continue the study of separable elements in finite Weyl groups, introduced in [11]. These elements generalize the well-studied class of separable permutations. We show that the multiplication map W/U×U→W is a length-additive bijection or splitting of the Weyl group W when U is an order ideal in right weak order generated by a separable element, where W/U denotes the generalized quotient. This generalizes a result for the symmetric group, answering an open problem of Wei [17].For a generalized quotient of the symmetric group, we show that this multiplication map is a bijection if and only if U is an order ideal in right weak order generated by a separable element, thereby classifying those generalized quotients which induce splittings of the symmetric group, resolving a problem of Björner and Wachs from 1988 [4]. We also prove that this map is always surjective when U is an order ideal in right weak order. Interpreting these sets of permutations as linear extensions of 2-dimensional posets gives the first direct combinatorial proof of an inequality due originally to Sidorenko in 1991, answering an open problem Morales, Pak, and Panova [13]. We also prove a new q-analog of Sidorenko's formula. All of these results are conjectured to extend to arbitrary finite Weyl groups.Finally, we show that separable elements in W are in bijection with the faces of all dimensions of several copies of the graph associahedron of the Dynkin diagram of W. This correspondence associates to each separable element w a certain nested set; we give product formulas for the rank generating functions of the principal upper and lower order ideals generated by w in terms of these nested sets, generalizing several known formulas.

Classical waqf, juristic analogy and framework of awqāf doctrines

Purpose This paper aims to analyse the Sharīʿah premises of waqf (Islamic endowment), followed by dilating on the nature of argumentation among the classical jurists on its rules and principles. The paper critically analyses the edifice of the applied juristic analogy of different early jurists in deriving various waqf doctrines. The objective of analysing the jurisprudential framework of waqf in its classical mould is to conceptualise the methods, mechanism and nature of juristic analogies in deriving the waqf principles. This analysis is critical to understand the scope of jurisprudential flexibility in modern awqāf. Design/methodology/approach The paper is an outcome of a library-based research. It uses the classical jurisprudential treatises of waqf with an aim to analyse the Sharīʿah basis of the institution, the premises of its key principles and the applied juristic analogy to derive the same. The paper covers the classical waqf books and treatises from the four Sunni schools of jurisprudence and uses a textual analysis method. Findings The paper finds that in its initial phase, the conceptual framework of waqf was not unanimously agreed by all jurists, rather its Sharīʿah permissibility remained critically disputed among them for a while. Though, the opinion of those jurists who approved the Sharīʿah-validity of waqf was to prevail in the later stage, disagreement persisted with reference to its necessary features and defining criteria. It is found that in the classical waqf literature, two most disputed aspects of waqf jurisprudence constituted the requirements for completion of a waqf and its ownership status. Research limitations/implications This study neither covers the historical contribution of waqf among the Muslim societies nor touches on the empirical aspects of modern waqf. Rather, the focus of the study is limited to analysing the classical jurisprudential discourse of waqf and distillation process of its rulings. Practical implications The objective of analysing the classical juristic discourse of waqf is to underline the premises of classical juristic analogy in determining the framework of fiqh al-awqāf (jurisprudence of waqf) in its classical permutations and to learn how to adopt a similar approach for deduction of new waqf rulings. Originality/value This paper adds original value to the body of waqf literature for analysing the classical waqf rulings distillation process along with examining the methods and mechanism of juristic analogy.

More Low Differential Uniformity Permutations over F22k with k Odd

Permutations with low differential uniformity, high algebraic degree, and high nonlinearity over F22k can be used as the substitution boxes for many block ciphers. In this paper, several classes of low differential uniformity permutations are constructed based on the method of choosing two permutations over F22k to get the desired permutations. The resulted low differential uniformity permutations have high algebraic degrees and nonlinearities simultaneously, which provide more choices for the substitution boxes. Moreover, some numerical examples are provided to show the efficacy of the theoretical results.

The set of non-uniquely ergodic d-IETs has Hausdorff codimension 1/2

We show that the set of not uniquely ergodic d-IETs with permutation in the Rauzy class of the hyperelliptic permutation has Hausdorff dimension $$d-\frac{3}{2} $$ [in the $$(d-1)$$ -dimension space of d-IETs] for $$d\ge 5$$ . For $$d=4$$ this was shown by Athreya–Chaika and for $$d\in \{2,3\}$$ the set is known to have dimension $$d-2$$ . This provides lower bounds on the Hausdorff dimension of non-weakly mixing IETs and, with input from Al-Saqban et al. (Exceptional directions for the Teichmuller geodesic flow and Hausdorff dimension, 2017. arXiv:1711.10542 ), identifies the Hausdorff dimension of non-weakly mixing IETs with permutation $$(d,d-1,\ldots ,2,1)$$ when d is odd.

On the boomerang uniformity of some permutation polynomials

The boomerang attack, introduced by Wagner in 1999, is a cryptanalysis technique against block ciphers based on differential cryptanalysis. In particular it takes into consideration two differentials, one for the upper part of the cipher and one for the lower part, and it exploits the dependency of these two differentials. At Eurocrypt’18, Cid et al. introduced a new tool, called the Boomerang Connectivity Table (BCT), that permits to simplify this analysis. Next, Boura and Canteaut introduced an important parameter for cryptographic S-boxes called boomerang uniformity, that is the maximum value in the BCT. Very recently, the boomerang uniformity of some classes of permutations (in particular quadratic functions) have been studied by Li, Qu, Sun and Li, and by Mesnager, Tang and Xiong. In this paper we further study the boomerang uniformity of some non-quadratic differentially 4-uniform functions. In particular, we consider the case of the Bracken-Leander cubic function and three classes of 4-uniform functions constructed by Li, Wang and Yu, obtained from modifying the inverse functions.

An inverse model‐based multiobjective estimation of distribution algorithm using Random‐Forest variable importance methods

AbstractMost existing methods of multiobjective estimation of distributed algorithms apply the estimation of distribution of the Pareto‐solution on the decision space during the search and little work has proposed on making a regression‐model for representing the final solution set. Some inverse‐model‐based approaches were reported, such as inversed‐model of multiobjective evolutionary algorithm (IM‐MOEA), where an inverse functional mapping from Pareto‐Front to Pareto‐solution is constructed on nondominated solutions based on Gaussian process and random grouping technique. But some of the effective inverse models, during this process, may be removed. This paper proposes an inversed‐model based on random forest framework. The main idea is to apply the process of random forest variable importance that determines some of the best assignment of decision variables (xn) to objective functions (fm) for constructing Gaussian process in inversed‐models that map all nondominated solutions from the objective space to the decision space. In this work, three approaches have been used: classical permutation, Naïve testing approach, and novel permutation variable importance. The proposed algorithm has been tested on the benchmark test suite for evolutionary algorithms [modified Deb K, Thiele L, Laumanns M, Zitzler E (DTLZ) and Walking Fish Group (WFG)] and indicates that the proposed method is a competitive and promising approach.

An energy-efficient bi-objective no-wait permutation flowshop scheduling problem to minimize total tardiness and total energy consumption

In manufacturing scheduling, sustainability concerns that raise from the service-oriented performance criteria have seldom been studied in the literature. This study aims to fill this gap in the literature by integrating the different energy consumption levels at the operational level. Since energy-efficient scheduling ideas have recently been increasing its popularity in industry due to the need for sustainable production, this study will be a good resource for future energy-efficient scheduling problems. Energy consumption in high volume manufacturing is a significant cost item in most industries. Potential energy saving mechanisms are needed to be integrated into manufacturing facilities for cost minimization at the operational level. A leading energy-saving mechanism in manufacturing is to be able to adapt/change the machine speed levels which exactly determines the energy consumption of the machines. Hence, in this study, the afore-mentioned framework is applied to the no-wait permutation flowshop scheduling problem (NWPFSP) which is a variant of classical permutation flowshop scheduling problems. However, it has various critical applications in industries such as chemical, pharmaceutical, food-processing, etc. This study proposes both mixed-integer linear programming (MILP) and constraint programming (CP) model formulations for the energy-efficient bi-objective no-wait permutation flowshop scheduling problems (NWPFSPs) considering the total tardiness and the total energy consumption minimization simultaneously. This problem treats total energy consumption as a second objective. Thus, the trade-off between the total tardiness – a service level measurement indicator – and the total energy consumption – a sustainability level indicator – is analyzed in this study. Furthermore, due to the NP-hardness nature of the first objective of the problem, a novel multi-objective discrete artificial bee colony algorithm (MO-DABC), a traditional multi-objective genetic algorithm (MO-GA) and a variant of multi-objective genetic algorithm with a local search (MO-GALS) are proposed for the bi-objective no-wait permutation flowshop scheduling problem. Besides the proposed algorithms are compared with the multi-objective energy-efficient algorithms from the literature. Consequently, a comprehensive comparative metaheuristic analysis is carried out. The computational results indicate that the proposed MO-DABC algorithm outperforms MILP, CP, MO-GA, MO-GALS, and algorithms from the literature in terms of both cardinality and quality of the solutions. The powerful results of this study show that the proposed models and algorithms can be adapted to other energy-efficient scheduling problems such as no-idle flowshop, blocking flowshop and job-shop scheduling problems or to other higher-level integrated manufacturing problems.

Permutations from an arithmetic setting

Let m,n be positive integers such that m>1 divides n. In this paper, we introduce a special class of piecewise-affine permutations of the finite set [1,n]≔{1,…,n} with the property that the reduction (modm) of m consecutive elements in any of its cycles is, up to a cyclic shift, a fixed permutation of [1,m]. Our main result provides the cycle decomposition of such permutations. We further show that such permutations give rise to permutations of finite fields. In particular, we explicitly obtain classes of permutation polynomials of finite fields whose cycle decomposition and its inverse are explicitly given.

A structural characterisation of Av(1324) and new bounds on its growth rate

We establish an improved lower bound of 10.271 for the exponential growth rate of the class of permutations avoiding the pattern 1324, and an improved upper bound of 13.5. These results depend on a new exact structural characterisation of 1324-avoiders as a subclass of an infinite staircase grid class, together with precise asymptotics of a small domino subclass whose enumeration we relate to West-two-stack-sortable permutations and planar maps. The bounds are established by carefully combining copies of the dominoes in particular ways consistent with the structural characterisation. The lower bound depends on concentration results concerning the substructure of a typical domino, the determination of exactly when dominoes can be combined in the fewest distinct ways, and technical analysis of the resulting generating function.

Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

The m-Tamari lattice of F. Bergeron is an analogue of the classical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices.In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)n−1, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.

Low-Complexity Nonlinear Self-Inverse Permutation for Creating Physically Clone-Resistant Identities

New large classes of permutations over ℤ 2 n based on T-Functions as Self-Inverting Permutation Functions (SIPFs) are presented. The presented classes exhibit negligible or low complexity when implemented in emerging FPGA technologies. The target use of such functions is in creating the so called Secret Unknown Ciphers (SUC) to serve as resilient Clone-Resistant structures in smart non-volatile Field Programmable Gate Arrays (FPGA) devices. SUCs concepts were proposed a decade ago as digital consistent alternatives to the conventional analog inconsistent Physical Unclonable Functions PUFs. The proposed permutation classes are designed and optimized particularly to use non-consumed Mathblock cores in programmable System-on-Chip (SoC) FPGA devices. Hardware and software complexities for realizing such structures are optimized and evaluated for a sample expected target FPGA technology. The attained security levels of the resulting SUCs are evaluated and shown to be scalable and usable even for post-quantum crypto systems.