Generation of the symmetric group Sn2

2021 ◽  
pp. 2150107
Author(s):  
Carlos Zequeira Sánchez ◽  
Evaristo José Madarro Capó ◽  
Guillermo Sosa-Gómez
Keyword(s):  
Symmetric Group ◽  
Random Element ◽  
Matrix Representation ◽  
Random Permutation ◽  
Random Permutations ◽  
Dimension Formula ◽  
Indispensable Tool

In various scenarios today, the generation of random permutations has become an indispensable tool. Since random permutation of dimension [Formula: see text] is a random element of the symmetric group [Formula: see text], it is necessary to have algorithms capable of generating any permutation. This work demonstrates that it is possible to generate the symmetric group [Formula: see text] by shifting the components of a particular matrix representation of each permutation.

scholarly journals Random permutations and their discrepancy process

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Computational Biology ◽  
Symmetric Group ◽  
Gaussian Process ◽  
Brownian Bridge ◽  
Random Permutation ◽  
Partial Sums ◽  
Random Permutations ◽  
New Process ◽  
International Audience

International audience Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.

scholarly journals Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

2021 ◽  
Author(s):  
Joseph Wilson ◽  
Matt Visser
Keyword(s):  
Efficient Method ◽  
Geometric Algebra ◽  
Clifford Algebras ◽  
Matrix Representation ◽  
Spin Representation ◽  
Lorentz Transformations ◽  
Rodrigues Formula ◽  
Dimension Formula ◽  
Spacetime Algebra ◽  
Geometric Algebras

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

scholarly journals Random permutations fix a worst case for cyclic coordinate descent

2018 ◽  
Vol 39 (3) ◽  
pp. 1246-1275 ◽  
Author(s):  
Ching-pei Lee ◽  
Stephen J Wright
Keyword(s):  
Nonlinear Function ◽  
Quadratic Function ◽  
Random Permutation ◽  
Coordinate Descent ◽  
Random Permutations ◽  
Worst Case ◽  
Computational Performance ◽  
Technical Report ◽  
Cyclic Coordinate Descent ◽  
Better Than

Abstract Variants of the coordinate descent approach for minimizing a nonlinear function are distinguished in part by the order in which coordinates are considered for relaxation. Three common orderings are cyclic (CCD), in which we cycle through the components of $x$ in order; randomized (RCD), in which the component to update is selected randomly and independently at each iteration; and random-permutations cyclic (RPCD), which differs from CCD only in that a random permutation is applied to the variables at the start of each cycle. Known convergence guarantees are weaker for CCD and RPCD than for RCD, though in most practical cases, computational performance is similar among all these variants. There is a certain type of quadratic function for which CCD is significantly slower than for RCD; a recent paper by Sun & Ye (2016, Worst-case complexity of cyclic coordinate descent: $O(n^2)$ gap with randomized version. Technical Report. Stanford, CA: Department of Management Science and Engineering, Stanford University. arXiv:1604.07130) has explored the poor behavior of CCD on functions of this type. The RPCD approach performs well on these functions, even better than RCD in a certain regime. This paper explains the good behavior of RPCD with a tight analysis.

APPLYING BUCKET RANDOM PERMUTATIONS TO STATIONARY SEQUENCES WITH LONG-RANGE DEPENDENCE

Fractals ◽  
2007 ◽  
Vol 15 (02) ◽  
pp. 105-126 ◽  
Author(s):  
YINGCHUN ZHOU ◽  
MURAD S. TAQQU
Keyword(s):  
Long Range ◽  
Short Range ◽  
Random Permutation ◽  
Stationary Sequence ◽  
Long Range Dependence ◽  
Finite Variance ◽  
Dependence Structure ◽  
Random Permutations ◽  
Medium Range ◽  
The Time Domain

Bucket random permutations (shuffling) are used to modify the dependence structure of a time series, and this may destroy long-range dependence, when it is present. Three types of bucket permutations are considered here: external, internal and two-level permutations. It is commonly believed that (1) an external random permutation destroys the long-range dependence and keeps the short-range dependence, (2) an internal permutation destroys the short-range dependence and keeps the long-range dependence, and (3) a two-level permutation distorts the medium-range dependence while keeping both the long-range and short-range dependence. This paper provides a theoretical basis for investigating these claims. It extends the study started in Ref. 1 and analyze the effects that these random permutations have on a long-range dependent finite variance stationary sequence both in the time domain and in the frequency domain.

scholarly journals On the semantic security of cellular automata based pseudo-random permutation using results from the Luby-Rackoff construction

2015 ◽  
Vol 15 (1) ◽  
pp. 21
Author(s):  
Kamel Mohammed Faraoun
Keyword(s):  
Cellular Automata ◽  
Block Cipher ◽  
Random Permutation ◽  
Block Ciphers ◽  
Random Permutations ◽  
Semantic Security ◽  
Transition Rules ◽  
Reversible Cellular Automata ◽  
Symmetric Block ◽  
Number Of Iterations

This paper proposes a semantically secure construction of pseudo-random permutations using second-order reversible cellular automata. We show that the proposed construction is equivalent to the Luby-Rackoff model if it is built using non-uniform transition rules, and we prove that the construction is strongly secure if an adequate number of iterations is performed. Moreover, a corresponding symmetric block cipher is constructed and analysed experimentally in comparison with popular ciphers. Obtained results approve robustness and efficacy of the construction, while achieved performances overcome those of some existing block ciphers.

A Generalization of the Erdős–Turán Law for the Order of Random Permutation

2012 ◽  
Vol 21 (5) ◽  
pp. 715-733 ◽  
Author(s):  
ALEXANDER GNEDIN ◽  
ALEXANDER IKSANOV ◽  
ALEXANDER MARYNYCH
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Random Permutation ◽  
Unit Interval ◽  
Random Permutations ◽  
Cycle Structure ◽  
Limit Laws ◽  
Cycle Lengths

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on n integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős–Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM(θ)-distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

scholarly journals Note on a paper of Tsuzuku

1964 ◽  
Vol 6 (4) ◽  
pp. 196-197
Author(s):  
H. K. Farahat
Keyword(s):  
Symmetric Group ◽  
Commutative Ring ◽  
Permutation Group ◽  
Matrix Representation ◽  
Invertible Element ◽  
Unit Element ◽  
Transitive Permutation Group ◽  
Natural Representation ◽  
Correct Proof ◽  
Permutation Matrices

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

scholarly journals Separation Probabilities for Products of Permutations

2013 ◽  
Vol 23 (2) ◽  
pp. 201-222 ◽  
Author(s):  
OLIVIER BERNARDI ◽  
ROSENA R. X. DU ◽  
ALEJANDRO H. MORALES ◽  
RICHARD P. STANLEY
Keyword(s):  
Exact Formula ◽  
Random Permutation ◽  
Random Permutations ◽  
Mixing Properties

We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements 1,2,. . .,k are in distinct cycles of the random permutation of {1,2,. . .,n} obtained as a product of two uniformly random n-cycles.

scholarly journals Patterns in Random Permutations Avoiding the Pattern 132

2016 ◽  
Vol 26 (1) ◽  
pp. 24-51 ◽  
Author(s):  
SVANTE JANSON
Keyword(s):  
Limit Distribution ◽  
Random Permutation ◽  
Random Permutations ◽  
Brownian Excursion

We consider a random permutation drawn from the set of 132-avoiding permutations of lengthnand show that the number of occurrences of another pattern σ has a limit distribution, after scaling bynλ(σ)/2, where λ(σ) is the length of σ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.

Quaternionic equiangular lines

2020 ◽  
Vol 20 (2) ◽  
pp. 273-284
Author(s):  
Boumediene Et-Taoui
Keyword(s):  
Symmetric Group ◽  
Matrix Representation ◽  
Complex Representation ◽  
Symmetry Groups ◽  
Complex Matrix ◽  
Equiangular Lines

AbstractLet 𝔽 = ℝ, ℂ or ℍ. A p-set of equi-isoclinic n-planes with parameter λ in 𝔽r is a set of pn-planes spanning 𝔽r each pair of which has the same non-zero angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$. It is known that via a complex matrix representation, a pair of isoclinic n-planes in ℍr with angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$ yields a pair of isoclinic 2n-planes in ℂ2r with angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$. In this article we characterize all the p-tuples of equi-isoclinic planes in ℂ2r which come via our complex representation from p-tuples of equiangular lines in ℍr. We then construct all the p-tuples of equi-isoclinic planes in ℂ4 and derive all the p-tuples of equiangular lines in ℍ2. Among other things it turns out that the quadruples of equiangular lines in ℍ2 are all regular, i.e. their symmetry groups are isomorphic to the symmetric group S4.

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