Fishburn permutations are in bijection with several important combinatorial structures including interval orders.In this paper,we use the method of generating trees to enumerate two classes of pattern-avoiding Fishbur...Fishburn permutations are in bijection with several important combinatorial structures including interval orders.In this paper,we use the method of generating trees to enumerate two classes of pattern-avoiding Fishburn permutations subject to 7 classical statistics simultaneously.The classes of our interest are(321,312)-avoiding and(321,4123)-avoiding Fishburn permutations.The statistics of our interest are ascents,descents,inversions,right-to-left maxima,right-to-left minima,left-to-right maxima and left-to-right minima.Our results generalize a result by Egge.展开更多
In combinatorics, permutations are important objects with many operations. In this paper, we define a coupling product on permutations and prove that the space spanned by permutations is a graded algebra.
In this work we propose efficient codec algorithms for watermarking images that are intended for uploading on the web under intellectual property protection. Headed to this direction, we recently suggested a way in wh...In this work we propose efficient codec algorithms for watermarking images that are intended for uploading on the web under intellectual property protection. Headed to this direction, we recently suggested a way in which an integer number w which being transformed into a self-inverting permutation, can be represented in a two dimensional (2D) object and thus, since images are 2D structures, we have proposed a watermarking algorithm that embeds marks on them using the 2D representation of w in the spatial domain. Based on the idea behind this technique, we now expand the usage of this concept by marking the image in the frequency domain. In particular, we propose a watermarking technique that also uses the 2D representation of self-inverting permutations and utilizes marking at specific areas thanks to partial modifications of the image’s Discrete Fourier Transform (DFT). Those modifications are made on the magnitude of specific frequency bands and they are the least possible additive information ensuring robustness and imperceptiveness. We have experimentally evaluated our algorithms using various images of different characteristics under JPEG compression. The experimental results show an improvement in comparison to the previously obtained results and they also depict the validity of our proposed codec algorithms.展开更多
Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that...Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a permutation a is a double derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double derangements with respect to x and y. Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, we show that if and z is a permutation such that for and for , then where .展开更多
Orthomorphic permutations have good characteristics in cryptosystems. In this paper, by using of knowledge about relation between orthomorphic permutations and multi-output functions, and conceptions of the generalize...Orthomorphic permutations have good characteristics in cryptosystems. In this paper, by using of knowledge about relation between orthomorphic permutations and multi-output functions, and conceptions of the generalized Walsh spectrum of multi-output functions and the auto-correlation function of multi-output functions to investigate the Walsh spectral characteristics and the auto-correlation function characteristics of orthormophic permutations, several results are obtained.展开更多
When solving a mathematical problem, we sometimes encounter a situation where we can not reach a correct answer in spite of acquiring knowledge and formula necessary for the solution. The reason can be attributed to t...When solving a mathematical problem, we sometimes encounter a situation where we can not reach a correct answer in spite of acquiring knowledge and formula necessary for the solution. The reason can be attributed to the lack in metacognitive abilities. Metacognitive abilities consist of comparing the difficulty of problem with own ability, proper plan of solution process, and conscious monitoring and control of solution process. The role and importance of metacognitive ability in mathematical problem solving of permutations and combinations was explored. Participants were required to solve five practical problems related to permutations and combinations. For each problem, the solution process was divided into： （1） understanding （recognition） of mathematical problem; （2） plan of solution; （3） execution of solution. Participants were also required to rate the anticipation whether they could solve it or not, and to rate the confidence of their own answer. According to the total score of five problems, the participants were categorized into the group of the high test score and the group of the low test score. As a result, at the plan and the execution processes, statistically significant differences were detected between the high and the low score groups. As for the rating on the anticipation of result and the confidence of own answer, no significant differences were found between both groups. Moreover, the relationship between the score of plan process and the score of execution process was statistically correlated. In other words, the more proper the plan process was conducted, the more proper solution the participants reached. In such a way, the importance of metacognitive ability in the solving process, especially the plan ability, was suggested.展开更多
Permutation polynomials is a hot topic in finite fields,they have many applications in different areas.Permutation binomials and trinomials over finite fields were studied recently.In thispaper,by using a powerful lem...Permutation polynomials is a hot topic in finite fields,they have many applications in different areas.Permutation binomials and trinomials over finite fields were studied recently.In thispaper,by using a powerful lemma given by Zieve and some degree 5 and 6 permutation polynomials over Fq,we construct somepermutation binomials over Fqm.展开更多
Linked partitions were introduced by Dykema(Dykema K J. Multilinear function series and transforms in free probability theory. Adv. Math., 2005, 208(1):351–407) in the study of the unsymmetrized T-transform in free p...Linked partitions were introduced by Dykema(Dykema K J. Multilinear function series and transforms in free probability theory. Adv. Math., 2005, 208(1):351–407) in the study of the unsymmetrized T-transform in free probability theory.Permutation is one of the most classical combinatorial structures. According to the linear representation of linked partitions, Chen et al.(Chen W Y C, Wu S Y J, Yan C H. Linked partitions and linked cycles. European J. Combin., 2008, 29(6): 1408–1426) de?ned the concept of singly covered minimal elements. Let L(n, k) denote the set of linked partitions of [n] with k singly covered minimal elements and let P(n, k) denote the set of permutations of [n] with k cycles. In this paper, we mainly establish two bijections between L(n, k) and P(n, k). The two bijections from a different perspective show the one-to-one correspondence between the singly covered minimal elements in L(n, k) and the cycles in P(n, k).展开更多
In this paper, a sufficient and necessary condition of quick trickle permutations is given from the point of inverse permutations. The bridge is built between quick trickle permutations and m-value logic functions. By...In this paper, a sufficient and necessary condition of quick trickle permutations is given from the point of inverse permutations. The bridge is built between quick trickle permutations and m-value logic functions. By the methods of the Chrestenson spectrum of m-value logic functions and the auto-correlation function of m-value logic functions to investigate the Chrestenson spectral characteristics and the auto-correlation function charac- teristics of inverse permutations of quick trickle permutations, a determinant arithmetic of quick trickle permutations is given. Using the results, it becomes easy to judge that a permutation is a quick trickle permutation or not by using computer. This gives a new pathway to study constructions and enumerations of quick trickle permutations.展开更多
Carlitz and Scoville(1974)introduced the polynomials An(x,y|α,β),which we refer to as the(α,β)-Eulerian polynomials.These polynomials count permutations based on Eulerian-Stirling statistics,including descents,asc...Carlitz and Scoville(1974)introduced the polynomials An(x,y|α,β),which we refer to as the(α,β)-Eulerian polynomials.These polynomials count permutations based on Eulerian-Stirling statistics,including descents,ascents,left-to-right maxima and right-to-left maxima.Carlitz and Scoville(1974)obtained the generating function for An(x,y|α,β).In this paper,we introduce a new family of polynomials,Pn(u,v,w,z|α,β),defined on descent-Stirling statistics of permutations including valleys,exterior peaks,right double descents,left double ascents,left-to-right maxima and right-to-left maxima.By employing the grammatical calculus introduced by Chen(1993),we establish a connection between the generating function for Pn(u,v,w,z|α,β)and the generating function for An(x,y|α,β).Using this connection,we derive the generating function for Pn(u,v,w,z|α,β),which can be specialized to obtain(α,β)-extensions of generating functions for peaks,left peaks,double ascents,right double ascents and left-right double ascents given by David and Barton(1962),Elizalde and Noy(2003),Entringer(1969),Gessel and Zhuang(2018),Kitaev(2007),and Zhuang(2016).Moreover,we establish two relations between Pn(u,v,w,z|α,β)and An(x,y|α,β),which enable us to derive(α,β)-extensions of results obtained by Stembridge(1997),Petersen(2006),Brändén(2008)and Zhuang(2017),respectively.We also establish the left peak version of Stembridge’s formula and the peak version of Petersen’s formula,along with their respective(α,β)-extensions,by utilizing these two relations.Specializing(α,β)-extensions of Stembridge’s formula and the left peak version of Stembridge’s formula allows us to derive(α,β)-extensions of the tangent and secant numbers.展开更多
In this article,we shall explore some techniques of permutations(排列)and combi- nations.This is going to be a slightly long ar- ticle,but is very much educative,and I'll try to cover the entire thing from front t...In this article,we shall explore some techniques of permutations(排列)and combi- nations.This is going to be a slightly long ar- ticle,but is very much educative,and I'll try to cover the entire thing from front to back. You can skip bits if you know them,but展开更多
The traditional approaches to false discovery rate(FDR)control in multiple hypothesis testing are usually based on the null distribution of a test statistic.However,all types of null distributions,including the theore...The traditional approaches to false discovery rate(FDR)control in multiple hypothesis testing are usually based on the null distribution of a test statistic.However,all types of null distributions,including the theoretical,permutation-based and empirical ones,have some inherent drawbacks.For example,the theoretical null might fail because of improper assumptions on the sample distribution.Here,we propose a null distributionfree approach to FDR control for multiple hypothesis testing in the case-control study.This approach,named target-decoy procedure,simply builds on the ordering of tests by some statistic or score,the null distribution of which is not required to be known.Competitive decoy tests are constructed from permutations of original samples and are used to estimate the false target discoveries.We prove that this approach controls the FDR when the score function is symmetric and the scores are independent between different tests.Simulation demonstrates that it is more stable and powerful than two popular traditional approaches,even in the existence of dependency.Evaluation is also made on two real datasets,including an arabidopsis genomics dataset and a COVID-19 proteomics dataset.展开更多
The traditional academic warning methods for students in higher vocational colleges are relatively backward,single,and have many influencing factors,which have a limited effect on improving their learning ability.A da...The traditional academic warning methods for students in higher vocational colleges are relatively backward,single,and have many influencing factors,which have a limited effect on improving their learning ability.A data set was established by collecting academic warning data of students in a certain university.The importance of the school,major,grade,and warning level for the students was analyzed using the Pearson correlation coefficient,random forest variable importance,and permutation importance.It was found that the characteristic of the major has a great impact on the academic warning level.Countermeasures such as dynamic adjustment of majors,reform of cognitive adaptation of courses,full-cycle academic support,and data-driven precise intervention were proposed to provide theoretical support and practical paths for universities to improve the efficiency of academic warning and enhance students’learning ability.展开更多
The distributed permutation flow shop scheduling problem(DPFSP)has received increasing attention in recent years.The iterated greedy algorithm(IGA)serves as a powerful optimizer for addressing such a problem because o...The distributed permutation flow shop scheduling problem(DPFSP)has received increasing attention in recent years.The iterated greedy algorithm(IGA)serves as a powerful optimizer for addressing such a problem because of its straightforward,single-solution evolution framework.However,a potential draw-back of IGA is the lack of utilization of historical information,which could lead to an imbalance between exploration and exploitation,especially in large-scale DPFSPs.As a consequence,this paper develops an IGA with memory and learning mechanisms(MLIGA)to efficiently solve the DPFSP targeted at the mini-malmakespan.InMLIGA,we incorporate a memory mechanism to make a more informed selection of the initial solution at each stage of the search,by extending,reconstructing,and reinforcing the information from previous solutions.In addition,we design a twolayer cooperative reinforcement learning approach to intelligently determine the key parameters of IGA and the operations of the memory mechanism.Meanwhile,to ensure that the experience generated by each perturbation operator is fully learned and to reduce the prior parameters of MLIGA,a probability curve-based acceptance criterion is proposed by combining a cube root function with custom rules.At last,a discrete adaptive learning rate is employed to enhance the stability of the memory and learningmechanisms.Complete ablation experiments are utilized to verify the effectiveness of the memory mechanism,and the results show that this mechanism is capable of improving the performance of IGA to a large extent.Furthermore,through comparative experiments involving MLIGA and five state-of-the-art algorithms on 720 benchmarks,we have discovered that MLI-GA demonstrates significant potential for solving large-scale DPFSPs.This indicates that MLIGA is well-suited for real-world distributed flow shop scheduling.展开更多
Secure computing paradigms impose new architectural challenges for general-purpose processors. Cryptographic processing is needed for secure communications, storage, and computations. We identify two categories of ope...Secure computing paradigms impose new architectural challenges for general-purpose processors. Cryptographic processing is needed for secure communications, storage, and computations. We identify two categories of operations in symmetric-key and public-key cryptographic algorithms that are not common in previous general-purpose workloads： advanced bit operations within a word and multi-word operations. We define MOMR （Multiple Operands Multiple Results） execution or datarich execution as a unified solution to both challenges. It allows arbitrary n-bit permutations to be achieved in one or two cycles, rather than O（n） cycles as in existing RISC processors. It also enables significant acceleration of multiword multiplications needed by public-key ciphers. We propose two implementations of MOMR： one employs only hardware changes while the other uses Instruction Set Architecture （ISA） support. We show that MOMR execution leverages available resources in typical multi-issue processors with minimal additional cost. Multi-issue processors enhanced with MOMR units provide additional speedup over standard multi-issue processors with the same datapath. MOMR is a general architectural solution for word-oriented processor architectures to incorporate datarich operations.展开更多
Ever since the famous Erd os-Ko-Rado theorem initiated the study of intersecting families of subsets,extremal problems regarding intersecting properties of families of various combinatorial objects have been extensive...Ever since the famous Erd os-Ko-Rado theorem initiated the study of intersecting families of subsets,extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated.Among them,studies about families of subsets,vector spaces and permutations are of particular concerns.Recently,we proposed a new quantitative intersection problem for families of subsets:For F([n]k),define its total intersection number as I(F)=ΣF1;F2∈F|F1∩F2|.Then,what is the structure of F when it has the maximal total intersection number among all the families in([n]k)with the same family size?In a recent paper,Kong and Ge(2020)studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes.In this paper,we consider the analogues of this problem for families of vector spaces and permutations.For certain ranges of family sizes,we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers.To some extent,these results determine the unique structure of the optimal family for some certain values of jFj and characterize the relationship between having the maximal total intersection number and being intersecting.Besides,we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.展开更多
A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)...A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)which avoid patternsσ_(1),σ_(2),...,σ_(r).The aim of this paper is to investigate when,for certain patternsσ_(1),σ_(2),...,σ_(r),S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.We prove that for any{σ_(1),σ_(2),...,σ_(r)}?S_3,if{σ_(1),σ_(2),...,σ_(r)}is sign-balanced except for{132,213,231,312},then S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.In addition,we give some results in the case of avoiding some patterns of length 4.展开更多
Purpose–High-speed turnouts are more complex in structure and thus may cause abnormal vibration of highspeed train car body,affecting driving safety and passenger riding experience.Therefore,it is necessary to analyz...Purpose–High-speed turnouts are more complex in structure and thus may cause abnormal vibration of highspeed train car body,affecting driving safety and passenger riding experience.Therefore,it is necessary to analyze the data characteristics of continuous hunting of high-speed trains passing through turnouts and propose a diagnostic method for engineering applications.Design/methodology/approach–First,Complete Ensemble Empirical Mode Decomposition with Adaptive Noise(CEEMDAN)is performed to determine the first characteristic component of the car body’s lateral acceleration.Then,the Short-Time Fourier Transform(STFT)is performed to calculate the marginal spectra.Finally,the presence of a continuous hunting problem is determined based on the results of the comparison calculations and diagnostic thresholds.To improve computational efficiency,permutation entropy(PE)is used as a fast indicator to identify turnouts with potential problems.Findings–Under continuous hunting conditions,the PE is less than 0.90;the ratio of the maximum peak value of the signal component to the original signal peak value exceeded 0.7,and there is an energy band in the STFT time-frequency map,which corresponds to a frequency distribution range of 1–2 Hz.Originality/value–The research results have revealed the lateral vibration characteristics of the high-speed train’s car body during continuous hunting when passing through turnouts.On this basis,an effective diagnostic method has been proposed.With a focus on practical engineering applications,a rapid screening index for identifying potential issues has been proposed,significantly enhancing the efficiency of diagnostic processes.展开更多
Constructing permutation polynomials is a hot topic in finite fields,and permutation polynomials have many applications in dif‐ferent areas.In this paper,by using monomials on the cosets of a subgroup to characterize...Constructing permutation polynomials is a hot topic in finite fields,and permutation polynomials have many applications in dif‐ferent areas.In this paper,by using monomials on the cosets of a subgroup to characterize the permutational property of rational functions onμq+1,we construct a class of permutation quadrinomials with the form f_(r,a,b,c,s,t,u)(x)=x^(r)(1+ax^(s(q-1))+bx^(t(q-1))+cxu(q-1))of F_(q^(2)).展开更多
Quantum algorithms offer more enhanced computational efficiency in comparison to their classical counterparts when solving specific tasks.In this study,we implement the quantum permutation algorithm utilizing a polar ...Quantum algorithms offer more enhanced computational efficiency in comparison to their classical counterparts when solving specific tasks.In this study,we implement the quantum permutation algorithm utilizing a polar molecule within an external electric field.The selection of the molecular qutrit involves the utilization of field-dressed states generated through the pendular modes of SrO.Through the application of multi-target optimal control theory,we strategically design microwave pulses to execute logical operations,including Fourier transform,oracle U_(f)operation,and inverse Fourier transform within a three-level molecular qutrit structure.The observed high fidelity of our outcomes is intricately linked to the concept of the quantum speed limit,which quantifies the maximum speed of quantum state manipulation.Subsequently,we design the optimized pulse sequence to successfully simulate the quantum permutation algorithm on a single SrO molecule,achieving remarkable fidelity.Consequently,a quantum circuit comprising a single qutrit suffices to determine permutation parity with just a single function evaluation.Therefore,our results indicate that the optimal control theory can be well applied to the quantum computation of polar molecular systems.展开更多
基金National Natural Science Foundation of China(Grant No.12171362)。
文摘Fishburn permutations are in bijection with several important combinatorial structures including interval orders.In this paper,we use the method of generating trees to enumerate two classes of pattern-avoiding Fishburn permutations subject to 7 classical statistics simultaneously.The classes of our interest are(321,312)-avoiding and(321,4123)-avoiding Fishburn permutations.The statistics of our interest are ascents,descents,inversions,right-to-left maxima,right-to-left minima,left-to-right maxima and left-to-right minima.Our results generalize a result by Egge.
文摘In combinatorics, permutations are important objects with many operations. In this paper, we define a coupling product on permutations and prove that the space spanned by permutations is a graded algebra.
文摘In this work we propose efficient codec algorithms for watermarking images that are intended for uploading on the web under intellectual property protection. Headed to this direction, we recently suggested a way in which an integer number w which being transformed into a self-inverting permutation, can be represented in a two dimensional (2D) object and thus, since images are 2D structures, we have proposed a watermarking algorithm that embeds marks on them using the 2D representation of w in the spatial domain. Based on the idea behind this technique, we now expand the usage of this concept by marking the image in the frequency domain. In particular, we propose a watermarking technique that also uses the 2D representation of self-inverting permutations and utilizes marking at specific areas thanks to partial modifications of the image’s Discrete Fourier Transform (DFT). Those modifications are made on the magnitude of specific frequency bands and they are the least possible additive information ensuring robustness and imperceptiveness. We have experimentally evaluated our algorithms using various images of different characteristics under JPEG compression. The experimental results show an improvement in comparison to the previously obtained results and they also depict the validity of our proposed codec algorithms.
文摘Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a permutation a is a double derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double derangements with respect to x and y. Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, we show that if and z is a permutation such that for and for , then where .
基金Supported by State Key Laboratory of InformationSecurity Opening Foundation(01-02) .
文摘Orthomorphic permutations have good characteristics in cryptosystems. In this paper, by using of knowledge about relation between orthomorphic permutations and multi-output functions, and conceptions of the generalized Walsh spectrum of multi-output functions and the auto-correlation function of multi-output functions to investigate the Walsh spectral characteristics and the auto-correlation function characteristics of orthormophic permutations, several results are obtained.
文摘When solving a mathematical problem, we sometimes encounter a situation where we can not reach a correct answer in spite of acquiring knowledge and formula necessary for the solution. The reason can be attributed to the lack in metacognitive abilities. Metacognitive abilities consist of comparing the difficulty of problem with own ability, proper plan of solution process, and conscious monitoring and control of solution process. The role and importance of metacognitive ability in mathematical problem solving of permutations and combinations was explored. Participants were required to solve five practical problems related to permutations and combinations. For each problem, the solution process was divided into： （1） understanding （recognition） of mathematical problem; （2） plan of solution; （3） execution of solution. Participants were also required to rate the anticipation whether they could solve it or not, and to rate the confidence of their own answer. According to the total score of five problems, the participants were categorized into the group of the high test score and the group of the low test score. As a result, at the plan and the execution processes, statistically significant differences were detected between the high and the low score groups. As for the rating on the anticipation of result and the confidence of own answer, no significant differences were found between both groups. Moreover, the relationship between the score of plan process and the score of execution process was statistically correlated. In other words, the more proper the plan process was conducted, the more proper solution the participants reached. In such a way, the importance of metacognitive ability in the solving process, especially the plan ability, was suggested.
基金Supported Partially by the National Natural Science Foundation of China(11926344)Science and Technology Research Projects of Chongqing Municipal Education Commission(KJQN201901402,KJQN201900506)Fund Project of Chongqing Normal University(17XWB021)。
文摘Permutation polynomials is a hot topic in finite fields,they have many applications in different areas.Permutation binomials and trinomials over finite fields were studied recently.In thispaper,by using a powerful lemma given by Zieve and some degree 5 and 6 permutation polynomials over Fq,we construct somepermutation binomials over Fqm.
基金The NSF(11601020,11501014) of China2017 Commercial Specialty Project(19005757053) of BTBU2018 Postgraduate Research Capacity Improvement Project(19008001491) of BTBU
文摘Linked partitions were introduced by Dykema(Dykema K J. Multilinear function series and transforms in free probability theory. Adv. Math., 2005, 208(1):351–407) in the study of the unsymmetrized T-transform in free probability theory.Permutation is one of the most classical combinatorial structures. According to the linear representation of linked partitions, Chen et al.(Chen W Y C, Wu S Y J, Yan C H. Linked partitions and linked cycles. European J. Combin., 2008, 29(6): 1408–1426) de?ned the concept of singly covered minimal elements. Let L(n, k) denote the set of linked partitions of [n] with k singly covered minimal elements and let P(n, k) denote the set of permutations of [n] with k cycles. In this paper, we mainly establish two bijections between L(n, k) and P(n, k). The two bijections from a different perspective show the one-to-one correspondence between the singly covered minimal elements in L(n, k) and the cycles in P(n, k).
基金the Opening Foundation of State Key Labo-ratory of Information Security (20050102)
文摘In this paper, a sufficient and necessary condition of quick trickle permutations is given from the point of inverse permutations. The bridge is built between quick trickle permutations and m-value logic functions. By the methods of the Chrestenson spectrum of m-value logic functions and the auto-correlation function of m-value logic functions to investigate the Chrestenson spectral characteristics and the auto-correlation function charac- teristics of inverse permutations of quick trickle permutations, a determinant arithmetic of quick trickle permutations is given. Using the results, it becomes easy to judge that a permutation is a quick trickle permutation or not by using computer. This gives a new pathway to study constructions and enumerations of quick trickle permutations.
基金supported by National Natural Science Foundation of China(Grant No.12171358).
文摘Carlitz and Scoville(1974)introduced the polynomials An(x,y|α,β),which we refer to as the(α,β)-Eulerian polynomials.These polynomials count permutations based on Eulerian-Stirling statistics,including descents,ascents,left-to-right maxima and right-to-left maxima.Carlitz and Scoville(1974)obtained the generating function for An(x,y|α,β).In this paper,we introduce a new family of polynomials,Pn(u,v,w,z|α,β),defined on descent-Stirling statistics of permutations including valleys,exterior peaks,right double descents,left double ascents,left-to-right maxima and right-to-left maxima.By employing the grammatical calculus introduced by Chen(1993),we establish a connection between the generating function for Pn(u,v,w,z|α,β)and the generating function for An(x,y|α,β).Using this connection,we derive the generating function for Pn(u,v,w,z|α,β),which can be specialized to obtain(α,β)-extensions of generating functions for peaks,left peaks,double ascents,right double ascents and left-right double ascents given by David and Barton(1962),Elizalde and Noy(2003),Entringer(1969),Gessel and Zhuang(2018),Kitaev(2007),and Zhuang(2016).Moreover,we establish two relations between Pn(u,v,w,z|α,β)and An(x,y|α,β),which enable us to derive(α,β)-extensions of results obtained by Stembridge(1997),Petersen(2006),Brändén(2008)and Zhuang(2017),respectively.We also establish the left peak version of Stembridge’s formula and the peak version of Petersen’s formula,along with their respective(α,β)-extensions,by utilizing these two relations.Specializing(α,β)-extensions of Stembridge’s formula and the left peak version of Stembridge’s formula allows us to derive(α,β)-extensions of the tangent and secant numbers.
文摘In this article,we shall explore some techniques of permutations(排列)and combi- nations.This is going to be a slightly long ar- ticle,but is very much educative,and I'll try to cover the entire thing from front to back. You can skip bits if you know them,but
基金supported by the National Key R&D Program of China(No.2018YFB0704304)the National Natural Science Foundation of China(Nos.32070668,62002231,61832003,61433014)the K.C.Wong Education Foundation。
文摘The traditional approaches to false discovery rate(FDR)control in multiple hypothesis testing are usually based on the null distribution of a test statistic.However,all types of null distributions,including the theoretical,permutation-based and empirical ones,have some inherent drawbacks.For example,the theoretical null might fail because of improper assumptions on the sample distribution.Here,we propose a null distributionfree approach to FDR control for multiple hypothesis testing in the case-control study.This approach,named target-decoy procedure,simply builds on the ordering of tests by some statistic or score,the null distribution of which is not required to be known.Competitive decoy tests are constructed from permutations of original samples and are used to estimate the false target discoveries.We prove that this approach controls the FDR when the score function is symmetric and the scores are independent between different tests.Simulation demonstrates that it is more stable and powerful than two popular traditional approaches,even in the existence of dependency.Evaluation is also made on two real datasets,including an arabidopsis genomics dataset and a COVID-19 proteomics dataset.
基金supported by the Basic Ability Improvement Project of Young and Middle-Aged Teachers in Colleges and Universities of Guangxi(2022KY1922,2021KY1938).
文摘The traditional academic warning methods for students in higher vocational colleges are relatively backward,single,and have many influencing factors,which have a limited effect on improving their learning ability.A data set was established by collecting academic warning data of students in a certain university.The importance of the school,major,grade,and warning level for the students was analyzed using the Pearson correlation coefficient,random forest variable importance,and permutation importance.It was found that the characteristic of the major has a great impact on the academic warning level.Countermeasures such as dynamic adjustment of majors,reform of cognitive adaptation of courses,full-cycle academic support,and data-driven precise intervention were proposed to provide theoretical support and practical paths for universities to improve the efficiency of academic warning and enhance students’learning ability.
基金supported in part by the National Key Research and Development Program of China under Grant No.2021YFF0901300in part by the National Natural Science Foundation of China under Grant Nos.62173076 and 72271048.
文摘The distributed permutation flow shop scheduling problem(DPFSP)has received increasing attention in recent years.The iterated greedy algorithm(IGA)serves as a powerful optimizer for addressing such a problem because of its straightforward,single-solution evolution framework.However,a potential draw-back of IGA is the lack of utilization of historical information,which could lead to an imbalance between exploration and exploitation,especially in large-scale DPFSPs.As a consequence,this paper develops an IGA with memory and learning mechanisms(MLIGA)to efficiently solve the DPFSP targeted at the mini-malmakespan.InMLIGA,we incorporate a memory mechanism to make a more informed selection of the initial solution at each stage of the search,by extending,reconstructing,and reinforcing the information from previous solutions.In addition,we design a twolayer cooperative reinforcement learning approach to intelligently determine the key parameters of IGA and the operations of the memory mechanism.Meanwhile,to ensure that the experience generated by each perturbation operator is fully learned and to reduce the prior parameters of MLIGA,a probability curve-based acceptance criterion is proposed by combining a cube root function with custom rules.At last,a discrete adaptive learning rate is employed to enhance the stability of the memory and learningmechanisms.Complete ablation experiments are utilized to verify the effectiveness of the memory mechanism,and the results show that this mechanism is capable of improving the performance of IGA to a large extent.Furthermore,through comparative experiments involving MLIGA and five state-of-the-art algorithms on 720 benchmarks,we have discovered that MLI-GA demonstrates significant potential for solving large-scale DPFSPs.This indicates that MLIGA is well-suited for real-world distributed flow shop scheduling.
文摘Secure computing paradigms impose new architectural challenges for general-purpose processors. Cryptographic processing is needed for secure communications, storage, and computations. We identify two categories of operations in symmetric-key and public-key cryptographic algorithms that are not common in previous general-purpose workloads： advanced bit operations within a word and multi-word operations. We define MOMR （Multiple Operands Multiple Results） execution or datarich execution as a unified solution to both challenges. It allows arbitrary n-bit permutations to be achieved in one or two cycles, rather than O（n） cycles as in existing RISC processors. It also enables significant acceleration of multiword multiplications needed by public-key ciphers. We propose two implementations of MOMR： one employs only hardware changes while the other uses Instruction Set Architecture （ISA） support. We show that MOMR execution leverages available resources in typical multi-issue processors with minimal additional cost. Multi-issue processors enhanced with MOMR units provide additional speedup over standard multi-issue processors with the same datapath. MOMR is a general architectural solution for word-oriented processor architectures to incorporate datarich operations.
基金supported by National Natural Science Foundation of China (Grant No. 11971325)National Key Research and Development Program of China (Grant Nos. 2020YFA0712100 and 2018YFA0704703)Beijing Scholars Program
文摘Ever since the famous Erd os-Ko-Rado theorem initiated the study of intersecting families of subsets,extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated.Among them,studies about families of subsets,vector spaces and permutations are of particular concerns.Recently,we proposed a new quantitative intersection problem for families of subsets:For F([n]k),define its total intersection number as I(F)=ΣF1;F2∈F|F1∩F2|.Then,what is the structure of F when it has the maximal total intersection number among all the families in([n]k)with the same family size?In a recent paper,Kong and Ge(2020)studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes.In this paper,we consider the analogues of this problem for families of vector spaces and permutations.For certain ranges of family sizes,we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers.To some extent,these results determine the unique structure of the optimal family for some certain values of jFj and characterize the relationship between having the maximal total intersection number and being intersecting.Besides,we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.
基金Supported by the National Natural Science Foundation of China(Grant No.12061030)the Natural Science Foundation of Hainan Province(Grant No.122RC652)2023 Excellent Science and Technology Innovation Team of Jiangsu Province Universities(Real-Time Industrial Internet of Things).
文摘A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations.Let S_(n)(σ_(1),σ_(2),...,σ_(r))denote the set of permutations in the symmetric group S_(n)which avoid patternsσ_(1),σ_(2),...,σ_(r).The aim of this paper is to investigate when,for certain patternsσ_(1),σ_(2),...,σ_(r),S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.We prove that for any{σ_(1),σ_(2),...,σ_(r)}?S_3,if{σ_(1),σ_(2),...,σ_(r)}is sign-balanced except for{132,213,231,312},then S_(n)(σ_(1),σ_(2),...,σ_(r))is sign-balanced for every integer n>1.In addition,we give some results in the case of avoiding some patterns of length 4.
基金support from the funds of National Natural Science Foundation of China(52308473)China Academy of Railway Science Corporation Limited(2022YJ192)are gratefully acknowledged。
文摘Purpose–High-speed turnouts are more complex in structure and thus may cause abnormal vibration of highspeed train car body,affecting driving safety and passenger riding experience.Therefore,it is necessary to analyze the data characteristics of continuous hunting of high-speed trains passing through turnouts and propose a diagnostic method for engineering applications.Design/methodology/approach–First,Complete Ensemble Empirical Mode Decomposition with Adaptive Noise(CEEMDAN)is performed to determine the first characteristic component of the car body’s lateral acceleration.Then,the Short-Time Fourier Transform(STFT)is performed to calculate the marginal spectra.Finally,the presence of a continuous hunting problem is determined based on the results of the comparison calculations and diagnostic thresholds.To improve computational efficiency,permutation entropy(PE)is used as a fast indicator to identify turnouts with potential problems.Findings–Under continuous hunting conditions,the PE is less than 0.90;the ratio of the maximum peak value of the signal component to the original signal peak value exceeded 0.7,and there is an energy band in the STFT time-frequency map,which corresponds to a frequency distribution range of 1–2 Hz.Originality/value–The research results have revealed the lateral vibration characteristics of the high-speed train’s car body during continuous hunting when passing through turnouts.On this basis,an effective diagnostic method has been proposed.With a focus on practical engineering applications,a rapid screening index for identifying potential issues has been proposed,significantly enhancing the efficiency of diagnostic processes.
基金Supported by the National Natural Science Foundation of China(11926344)Science and Technology Research Program of Chongqing Municipal Education Commission(KJZD-K202401601)+1 种基金Doctor Talent Program of Chongqing University of Education(2023BSRC003)Undergraduate Science Research Program of Chongqing University of Education(KY20240046)。
文摘Constructing permutation polynomials is a hot topic in finite fields,and permutation polynomials have many applications in dif‐ferent areas.In this paper,by using monomials on the cosets of a subgroup to characterize the permutational property of rational functions onμq+1,we construct a class of permutation quadrinomials with the form f_(r,a,b,c,s,t,u)(x)=x^(r)(1+ax^(s(q-1))+bx^(t(q-1))+cxu(q-1))of F_(q^(2)).
基金supported by the National Natural Science Foundation of China under Grant Nos.92265209,11174081 and 62305285the Natural Science Foundation of Chongqing under Grant No.CSTB2024NSCQ-MSX0643the Shanghai Municipal Science and Technology Major Project under Grant No.2019SHZDZX01。
文摘Quantum algorithms offer more enhanced computational efficiency in comparison to their classical counterparts when solving specific tasks.In this study,we implement the quantum permutation algorithm utilizing a polar molecule within an external electric field.The selection of the molecular qutrit involves the utilization of field-dressed states generated through the pendular modes of SrO.Through the application of multi-target optimal control theory,we strategically design microwave pulses to execute logical operations,including Fourier transform,oracle U_(f)operation,and inverse Fourier transform within a three-level molecular qutrit structure.The observed high fidelity of our outcomes is intricately linked to the concept of the quantum speed limit,which quantifies the maximum speed of quantum state manipulation.Subsequently,we design the optimized pulse sequence to successfully simulate the quantum permutation algorithm on a single SrO molecule,achieving remarkable fidelity.Consequently,a quantum circuit comprising a single qutrit suffices to determine permutation parity with just a single function evaluation.Therefore,our results indicate that the optimal control theory can be well applied to the quantum computation of polar molecular systems.
