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: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.

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 Local extrema in random permutations and the structure of longest alternating subsequences

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Dan Romik
Keyword(s):  
Joint Distribution ◽  
Random Permutation ◽  
Random Permutations ◽  
Local Minima ◽  
New Approach ◽  
Successive Minimum ◽  
Local Extrema ◽  
Nous Montrons ◽  
Dimensional Distribution ◽  
International Audience

International audience Let $\textbf{as}_n$ denote the length of a longest alternating subsequence in a uniformly random permutation of order $n$. Stanley studied the distribution of $\textbf{as}_n$ using algebraic methods, and showed in particular that $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ and $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. From Stanley's result it can be shown that after rescaling, $\textbf{as}_n$ converges in the limit to the Gaussian distribution. In this extended abstract we present a new approach to the study of $\textbf{as}_n$ by relating it to the sequence of local extrema of a random permutation, which is shown to form a "canonical'' longest alternating subsequence. Using this connection we reprove the abovementioned results in a more probabilistic and transparent way. We also study the distribution of the values of the local minima and maxima, and prove that in the limit the joint distribution of successive minimum-maximum pairs converges to the two-dimensional distribution whose density function is given by $f(s,t) = 3(1-s)t e^{t-s}$. Pour une permutation aléatoire d'ordre $n$, on désigne par $\textbf{as}_n$ la longueur maximale d'une de ses sous-suites alternantes. Stanley a étudié la distribution de $\textbf{as}_n$ en utilisant des méthodes algébriques, et il a démontré en particulier que $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ et $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. A partir du résultat de Stanley on peut montrer qu'après changement d'échelle, $\textbf{as}_n$ converge vers la distribution normale. Nous présentons ici une approche nouvelle pour l'étude de $\textbf{as}_n$, en la reliant à la suite des extrema locaux d'une permutation aléatoire, dont nous montrons qu'elle constitue une sous-suite alternante maximale "canonique''. En utilisant cette relation, nous prouvons à nouveau les résultats mentionnés ci-dessus d'une façon plus probabiliste et transparente. En plus, nous prouvons un résultat asymptotique sur la distribution limite des paires formées d'un minimum et d'un maximum locaux consécutifs.

scholarly journals Bridges and random truncations of random matrices

2014 ◽  
Vol 03 (02) ◽  
pp. 1450006 ◽  
Author(s):  
Vincent Beffara ◽  
Catherine Donati-Martin ◽  
Alain Rouault
Keyword(s):  
Gaussian Process ◽  
Random Matrices ◽  
Brownian Bridge ◽  
Two Parameter ◽  
The Way

Let U be a Haar distributed matrix in 𝕌(n) or 𝕆(n). In a previous paper, we proved that after centering, the two-parameter process [Formula: see text] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of U by a random one, in which each row (respectively, column) is chosen with probability s (respectively, t) independently. We prove that the corresponding two-parameter process, after centering and normalization by n-1/2 converges to a Gaussian process. On the way we meet other interesting convergences.

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.

scholarly journals Minimal factorizations of a cycle: a multivariate generating function

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Simple Formula ◽  
Hook Length ◽  
Noncrossing Partitions ◽  
Long Cycle ◽  
Number Of Factors ◽  
Multivariate Generalization ◽  
International Audience

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

scholarly journals Intervals and factors in the Bruhat order

2015 ◽  
Vol Vol. 17 no. 1 (Combinatorics) ◽  
Author(s):  
Bridget Eileen Tenner
Keyword(s):  
Symmetric Group ◽  
Bruhat Order ◽  
Order Ideal ◽  
Principal Order ◽  
International Audience ◽  
Reduced Words

Combinatorics International audience In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related by a certain property of their reduced words. We show that the property does not hold when w is a decomposable permutation, and that the property always holds when w is the longest permutation.

scholarly journals Equivalences for pattern avoiding involutions and classification

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Mark Dukes ◽  
Vít Jelínek ◽  
Toufik Mansour ◽  
Astrid Reifegerste
Keyword(s):  
Symmetric Group ◽  
Signed Permutations ◽  
International Audience

International audience We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of $S_5$, $S_6$, and $S_7$ for both permutations and involutions. Nous complétons la classification de Wilf des motifs signés de longueur 5 à la fois pour les permutations signées et les involutions signées. Nous donnons de nouvelles équivalences générales de motifs qui prouvent les conjectures de Jaggard concernant les involutions dans le groupe symétrique évitant certains motifs de longueur 5 et 6. De cette manière nous complétons également la classification de Wilf de $S_5$, $S_6$ et $S_7$ à la fois pour les permutations et les involutions.

scholarly journals A phase transition in the random transposition random walk

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Nathanael Berestycki ◽  
Rick Durrett
Keyword(s):  
Phase Transition ◽  
Genome Rearrangement ◽  
Graph Model ◽  
Random Permutation ◽  
Surprising Result ◽  
Parsimony Method ◽  
Random Graph Model ◽  
Minimum Number ◽  
International Audience ◽  
Random Transposition

International audience Our work is motivated by Bourque-Pevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 < c ≤ 1$, the distance $D_cn/2 ~ cn/2$, i.e., the distance increases linearly with time; for $c > 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Rényi random graph model.

Sign in / Sign up

Export Citation Format

Share Document