On Random Generation of the Symmetric Group

1993 ◽  
Vol 2 (4) ◽  
pp. 505-512 ◽  
Author(s):  
Tomasz Łuczak ◽  
László Pyber
Keyword(s):  
Symmetric Group ◽  
Random Permutation ◽  
Invariant Subset ◽  
Random Generation ◽  
Element Set

We prove that the probability i(n, k) that a random permutation of an n element set has an invariant subset of precisely k elements decreases as a power of k, for k ≤ n/2. Using this fact, we prove that the fraction of elements of Sn belong to transitive subgroups other than Sn or An tends to 0 when n → ∞, as conjectured by Cameron. Finally, we show that for every ∈ > 0 there exists a constant C such that C elements of the symmetric group Sn, chosen randomly and independently, generate invariably Sn with probability at least 1 − ∈. This confirms a conjecture of McKay.

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 Stability Analysis for $k$-wise Intersecting Families

10.37236/602 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vikram Kamat
Keyword(s):  
Stability Analysis ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Circle Method ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Element Set

We consider the following generalization of the seminal Erdős–Ko–Rado theorem, due to Frankl. For some $k\geq 2$, let $\mathcal{F}$ be a $k$-wise intersecting family of $r$-subsets of an $n$ element set $X$, i.e. for any $F_1,\ldots,F_k\in \mathcal{F}$, $\cap_{i=1}^k F_i\neq \emptyset$. If $r\leq \dfrac{(k-1)n}{k}$, then $|\mathcal{F}|\leq {n-1 \choose r-1}$. We prove a stability version of this theorem, analogous to similar results of Dinur-Friedgut, Keevash-Mubayi and others for the EKR theorem. The technique we use is a generalization of Katona's circle method, initially employed by Keevash, which uses expansion properties of a particular Cayley graph of the symmetric group.

scholarly journals The Inequalities of Merris and Foregger for Permanents

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1782
Author(s):  
Divya K. Udayan ◽  
Kanagasabapathi Somasundaram
Keyword(s):  
Symmetric Group ◽  
Linear Algebra ◽  
Symmetric Function ◽  
Stochastic Matrix ◽  
Elementary Symmetric Function ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Classes Of Matrices ◽  
The Matrix ◽  
Element Set

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

scholarly journals The Degree of the Splitting Field of a Random Polynomial over a Finite Field

10.37236/1823 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
John D. Dixon ◽  
Daniel Panario
Keyword(s):  
Finite Field ◽  
Symmetric Group ◽  
Random Permutation ◽  
Splitting Field ◽  
Random Polynomial ◽  
The Mean ◽  
Order Of An Element

The asymptotics of the order of a random permutation have been widely studied. P. Erdös and P. Turán proved that asymptotically the distribution of the logarithm of the order of an element in the symmetric group $S_{n}$ is normal with mean ${1\over2}(\log n)^{2}$ and variance ${1\over3}(\log n)^{3}$. More recently R. Stong has shown that the mean of the order is asymptotically $\exp(C\sqrt{n/\log n}+O(\sqrt{n}\log\log n/\log n))$ where $C=2.99047\ldots$. We prove similar results for the asymptotics of the degree of the splitting field of a random polynomial of degree $n$ over a finite field.

scholarly journals Asymptotic distributions of the number of restricted cycles in a random permutation

2009 ◽  
Vol 50 ◽  
Author(s):  
Tatjana Kargina
Keyword(s):  
Symmetric Group ◽  
Sufficient Conditions ◽  
Random Permutation ◽  
Value Distribution ◽  
Asymptotic Distributions ◽  
Necessary And Sufficient Conditions ◽  
Additive Functions ◽  
Number Of Cycles ◽  
Necessary And Sufficient

The value distribution of additive functions defined on the symmetric group with respect to the Ewens probability is examined. For the number of cycles with restricted lengths, we establish necessary and sufficient conditions under which the distributions converge weakly to a limit law.

scholarly journals On the Limiting Distribution for the Length of the Longest Alternating Sequence in a Random Permutation

10.37236/1051 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Harold Widom
Keyword(s):  
Symmetric Group ◽  
Explicit Formula ◽  
Generating Function ◽  
Random Permutation ◽  
Limiting Distribution

Recently Richard Stanley initiated a study of the distribution of the length as$_n(w)$ of the longest alternating subsequence in a random permutation $w$ from the symmetric group ${\cal S}_n$. Among other things he found an explicit formula for the generating function (on $n$ and $k$) for Pr$\,$(as$_n(w)\le k)$ and conjectured that the distribution, suitably centered and normalized, tended to a Gaussian with variance 8/45. In this note we present a proof of the conjecture based on the generating function.

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