Related papers: Thue-Morse at Multiples of an Integer

A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this…

For $H \ge N^{1-\frac{1}{2c}} \ln^2 N$, where $c$ is a fixed non-integer number satisfying $$ \|c\| \ge 3c\left(2^{[c]+1}-1\right)\frac{\ln \ln N}{\ln N}, \qquad c > \frac{4}{3}\left(1 + \frac{52\ln \ln N}{\ln N}\right), $$ we obtain an…

For a nonnegative integer $t$, let $c_t$ be the asymptotic density of natural numbers $n$ for which $s(n + t) \geq s(n)$, where $s(n)$ denotes the sum of digits of $n$ in base $2$. We prove that $c_t > 1/2$ for $t$ in a set of asymptotic…

Using three examples of sequences over a finite alphabet, we want to draw attention to the fact that these sequences having the minimum critical exponent in a given class of sequences show a large degree of symmetry, i.e., they are G-rich…

Univoque numbers are real numbers $\lambda > 1$ such that the number 1 admits a unique expansion in base $\lambda$, i.e., a unique expansion $1 = \sum_{j \geq 0} a_j \lambda^{-(j+1)}$, with $a_j \in \{0, 1, ..., \lceil \lambda \rceil -1\}$…

We use transference principle to show that whenever $s$ is suitably large depending on $k \geq 2$, every sufficiently large natural number $n$ satisfying some congruence conditions can be written in the form $n = p_1^k + \dots + p_s^k$,…

Let s_1,...,s_d be d positive integers and consider the multiple Hurwitz-zeta value zeta(s_1,...,s_d;-1/2,...,-1/2)/2^w where w=s_1+...+s_d is called the weight. For d<n+1, let T(2n,d) be the sum of all these values with even arguments…

Let T_k^m={\sigma \in S_k | \sigma_1=m}. We prove that the number of permutations which avoid all patterns in T_k^m equals (k-2)!(k-1)^{n+1-k} for k <= n. We then prove that for any \tau in T_k^1 (or any \tau in T_k^k), the number of…

Let \{X_1, X_2, ...\} be a sequence of positive independent and identically distributed random variables of Pareto-type with index \alpha>0 and let \{N(t); t\geq 0\} be a mixed Poisson process independent of the X_i's. For t\geq 0, define…

Let $S_n$ denote the symmetric group of order $n$. Say that two subsets $x, y\subseteq S_n$ are \emph{equivalent} if there exist permutations $g_1, g_2\in S_n$ such that $g_1xg_2=y$, where multiplication is understood elementwise. Recently,…

Let $\Sigma_{2n}$ be the set of all partitions of the even integers from the interval $(4,2n], n>2,$ into two odd prime parts. We select a partition from the set $\Sigma_{2n}$ uniformly at random. Let $2G_n$ be the number partitioned by…

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on…

Simon's congruence, denoted \sim_n, relates words having the same subwords of length up to n. We show that, over a k-letter alphabet, the number of words modulo \sim_n is in 2^{\Theta(n^{k-1} log n)}.

In this paper, we characterize the average Hamming weight distribution of subsequences of maximum-length sequences ($m$-sequences). In particular, we consider all possible $m$-sequences of dimension $k$ and find the average number of…

Given a finite set of bases $b_1$, $b_2$, \dots, $b_r$ (integers greater than $1$), a multi-base representation of an integer~$n$ is a sum with summands $db_1^{\alpha_1}b_2^{\alpha_2} \cdots b_r^{\alpha_r}$, where the $\alpha_j$ are…

The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) exactly $k$ times. Our motivation comes from the analogous study of pattern…

We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high…

For a positive integer $n$, the full transformation semigroup $T_n$ consists of all self maps of the set $\{1,\ldots,n\}$ under composition. Any finite semigroup $S$ embeds in some $T_n$, and the least such $n$ is called the (minimum…

Let K_n denote the smaller mode of the nth row of Stirling numbers of the second kind S(n, k). Using a probablistic argument, it is shown that for all n>=2, [exp(w(n))]-2<=K_n<=[exp(w(n))]+1, where [x] denotes the integer part of x, and…

We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size $n,$ decomposes into a sum of tensor…