Related papers: Balanced Stick Breaking

Let $\{u_n\}_n$ be a non-degenerate linear recurrence sequence of integers with Binet's formula given by $u_n= \sum_{i=1}^{m} P_i(n)\alpha_i^n.$ Assume $\max_i \vert \alpha_i \vert >1$. In 1977, Loxton and Van der Poorten conjectured that…

Inspired by the basic stick fragmentation model proposed by Becker et al. in arXiv:1309.5603v4, we consider three new versions of such fragmentation models, namely, continuous with random number of parts, continuous with probabilistic…

The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…

This paper gives a simple proof of a limit theorem for the lenght of the largest interval straddling a fixed number of i.i.d. points uniformly disributed on a unit interval. The key step in our argument is a classical theorem of Watson…

We study problems on covering $[0,1)$ by shrinking intervals centered at the points $\{q_n x\}$, where $(q_n)_{n\in \mathbb{N}}$ is a given real-valued sequence and $x \in [0,1)$ is random. For real-valued lacunary sequences…

It was shown by Gruslys, Leader and Tan that any finite subset of $\mathbb{Z}^n$ tiles $\mathbb{Z}^d$ for some $d$. The first non-trivial case is the punctured interval, which consists of the interval $\{-k,\ldots,k\} \subset \mathbb{Z}$…

In 2022, Gao, Huo, Liu, and Ma proved that every graph with minimum degree at least $k+1$ contains $k$ admissible cycles, where a set of $k$ cycles is said to be admissible if their lengths form an arithmetic progression with common…

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a…

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…

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are…

According to a classical result of Spencer, Szemer\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\H{o}s's lower bound, $n^{1+O(1/\log\log…

Let $C$ be the unit circle in $\mathbb{R}^2$. We can view $C$ as a plane graph whose vertices are all the points on $C$, and the distance between any two points on $C$ is the length of the smaller arc between them. We consider a graph…

A well-known result of Verstra\"ete \cite{V00} shows that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is at most twice the radius of…

A balanced generalized de Bruijn sequence with parameters $(n,l,k)$ is a cyclic sequence of $n$ bits such that (a) the number of 0's equals the number of 1's, and (b) each substring of length $l$ occurs at most $k$ times. We determine…

We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning…

The sequence reconstruction problem involves a model where a sequence is transmitted over several identical channels. This model investigates the minimum number of channels required for the unique reconstruction of the transmitted sequence.…

Fix integers $n \ge r \ge 2$. A clique partition of ${[n] \choose r}$ is a collection of proper subsets $A_1, A_2, ..., A_t \subset [n]$ such that $\bigcup_i{A_i \choose r}$ is a partition of ${[n] \choose r}$. Clique partitions are related…

The sequence starts with a(1) = 1; to extend it one writes the sequence so far as XY^k, where X and Y are strings of integers, Y is nonempty and k is as large as possible: then the next term is k. The sequence begins 1, 1, 2, 1, 1, 2, 2, 2,…

A de Bruijn covering code is a q-ary string S so that every q-ary string is at most R symbol changes from some n-word appearing consecutively in S. We introduce these codes and prove that they can have length close to the smallest possible…

For the random interval partition of $[0,1]$ generated by the uniform stick-breaking scheme known as GEM$(1)$, let $u_k$ be the probability that the first $k$ intervals created by the stick-breaking scheme are also the first $k$ intervals…