Related papers: Statistics for $S_n$ acting on $k$-sets

Let $S$ be a finite set, and $X_1,\ldots,X_n$ an i.i.d. uniform sample from $S$. To estimate the size $|S|$, without further structure, one can wait for repeats and use the birthday problem. This requires a sample size of the order…

In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq…

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a…

The statistical model assuming chemical equilibriumand local strangeness conservation describes most of the observed features of strange particle production from SIS up to RHIC. Deviations are found as the maximum in the measured K+/pi+…

For a fixed positive integer $\;k,\;$ limit laws of linearly normalized $\;k$-th upper order statistics are well known. In this article, a comprehensive study of tail behaviours of limit laws of normalized $k$-th upper order statistics…

We characterize all (absolute) 1-Lipschitz retracts Q of R^n with the maximum norm. Omitting two technical details, they coincide with the subsets written as the solution set of (at most) 2n inequalities like follows. For every coordinate…

Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect…

The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008)…

A base for a permutation group $G$ acting on a set $\Omega$ is a sequence $\mathcal{B}$ of points of $\Omega$ such that the pointwise stabiliser $G_{\mathcal{B}}$ is trivial. The base size of $G$ is the size of a smallest base for $G$. We…

Let $h:[0,1]\to\mathbb{R}$ be $C^2$ and such that $\sup_{[0,1]} h''<0$. For a (large) positive integer $n$, set $h_n(k) = n h(k/n)$ for any $k\in\{0,\dots,n\}$. We consider a random walk $(S_k)_{k\geq 0}$ with i.i.d.\ centred increments…

This paper discusses a general and useful stability principle which, roughly speaking, says that given a uniformly continuous function defined on an arbitrary metric space, if the function is bounded on the constraint set and we slightly…

The effects of dissipation on the scaling properties of nonlinear discontinuous maps are investigated by analyzing the behavior of the average squared action $\left< I^2 \right>$ as a function of the $n$-th iteration of the map as well as…

Let $(X, \mathcal{B},\mu,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $\epsilon>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \left\{ n\in\mathbb{N}\colon\mu(A\cap…

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that the sumset $NA$ has predictable size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in some finite cone other than all…

Let $r: S\times S\to \bb R_+$ be the jump rates of an irreducible random walk on a finite set $S$, reversible with respect to some probability measure $m$. For $\alpha >1$, let $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) =…

A Sidon set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d$. We improve on the lower bound on the diameter of a Sidon set with $k$ elements: if $k$ is sufficiently large and ${\cal A}$ is a Sidon set with…

Whiston proved that the maximum size of an irredundant generating set in the symmetric group $S_n$ is $n-1$, and Cameron and Cara characterized all irredundant generating sets of $S_n$ that achieve this size. Our goal is to extend their…

Let S_i be a random walk with standard exponential increments. We call \sum_{i=1}^k S_i its k-step area. The random variable V = \inf_{k \ge 1} \frac{2}{k(k+1)} \sum_{i=1}^k S_i plays important role in the study of so-called one-dimensional…

We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number \gamma, of dominating sets of size \gamma. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all…

Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer $n$ as a sum $n=\sum_k \epsilon_k U_k$, where the digits $\epsilon_k$ are taken from a finite alphabet $\Sigma$ and…