Related papers: Cycles in Mallows random permutations

Let $S_{\rm lcm}(n)$ denote the set of permutations $\pi$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,\pi(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $\pi$ of $[n]$ such that…

Suppose $\pi$ and $\pi'$ are two random elements of $S_n$ with constrained cycle types such that $\pi$ has $x n^{1/2}$ fixed points and $yn/2$ two-cycles, and likewise $\pi'$ has $x' n^{1/2}$ fixed points and $y'n/2$ two-cycles. We show…

Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $(x_1,\dots,x_n) \in[0,1]^n$ \textit{cyclic} if there exist independent random variables $U_1,\dots, U_n$ with $P(U_i=U_j)=0$ for $i\not=j$ such that…

In this text, we consider random permutations which can be written as free words in several independent random permutations: firstly, we fix a non trivial word $w$ in letters $g_1,g_1^{-1},..., g_k,g_k^{-1}$, secondly, for all $n$, we…

Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as $$\mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le…

In the first part of the paper, we study the inversion statistic of random permutations under the family $(\mathbb{P}_\theta^{(n)})_{\theta \ge 0}$ of Ewens sampling distributions on $S_n$. We obtain a rather simple exact formula for the…

Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the…

Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping $\phi\colon T_{n,m}\to T_{n,m}$ by $\phi(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability. We study the orbit structure of such…

Let $\pi$ be a cycle permutation that can be expressed as one-line $\pi = \pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and a cycle form $\pi = (c_1,c_2, ..., c_n)$. Archer et al. introduced the notion of pattern avoidance of one-line and all cycle…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$,…

Let $\pi$ be a cyclic permutation that can be expressed in its one-line form as $\pi = \pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and in its standard cycle form as $\pi = (c_1,c_2, ..., c_n)$ where $c_1=1$. Archer et al. introduced the notion of…

We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the…

Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. Next,…

The generalised random graph contains $n$ vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments and…

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in…

We count the number of occurrences of restricted patterns of length 3 in permutations with respect to length and the number of cycles. The main tool is a bijection between permutations in standard cycle form and weighted Motzkin paths.

In this paper, we study permutations $\pi \in S_n$ with exactly $m$ transpositions. In particular, we are interested in the expected value of $\pi(1)$ when such permutations are chosen uniformly at random. When $n$ is even, this expected…

We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the…

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…

We study the cycle structure of words in several random permutations. We assume that the permutations are independent and that their distribution is conjugation invariant, with a good control on their short cycles. If, after successive…