相关论文: On maximal repetitions of arbitrary exponent

Let $(\mathcal{P},\leqslant)$ be a finite poset. Define the numbers $a_1,a_2,\ldots$ (respectively, $c_1,c_2,\ldots$) so that $a_1+\ldots+a_k$ (respectively, $c_1+\ldots+c_k$) is the maximal number of elements of $\mathcal{P}$ which may be…

Given a positive rational number $n/d$ with $d$ odd, its odd greedy expansion starts with the largest odd denominator unit fraction at most $n/d$, adds the largest odd denominator unit fraction so the sum is at most $n/d$, and continues as…

We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the…

An arithmetic formula is an expression involving only the constant $1$, and the binary operations of addition and multiplication, with multiplication by $1$ not allowed. We obtain an asymptotic formula for the number of arithmetic formulas…

We consider systems of word equations and their solution sets. We discuss some fascinating properties of those, namely the size of a maximal independent set of word equations, and proper chains of solution sets of those. We recall the basic…

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 extension of a closure system on a finite set S as a closure system on the same set S containing the given one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set…

We study a new type of sequences whose elements are defined in terms of the position, sign and magnitude of another element of the sequence. The name ultra-recursive comes from the fact that these sequences possess terms that are generated…

Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…

Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit ball $\mathbb{B}_d$ is…

Let $r \ge 2$ and $s \ge 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and of the $s$-ary expansion of an irrational real number, viewed as infinite…

We address a question and a conjecture on the expected length of the longest common subsequences of two i.i.d.$\ $random permutations of $[n]:=\{1,2,...,n\}$. The question is resolved by showing that the minimal expectation is not attained…

In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare…

We investigate decomposable combinatorial labeled structures more fully, focusing on the exp-log class of type a=1 or 1/2. For instance, the modal length of the second longest cycle in a random n-permutation is (0.2350...)n, whereas the…

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several…

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…

We present a sharp upper bound for the number of generators of a finite group in terms of the ratio between the order and the exponent.

In [A. Frid, S. Puzynina, L.Q. Zamboni, \textit{On palindromic factorization of words}, Adv. in Appl. Math. 50 (2013), 737-748], it was conjectured that any infinite word whose palindromic lengths of factors are bounded is ultimately…

It is known that there are infinite words over finite alphabets with Abelian repetition threshold arbitrarily close to 1; however, the construction previously used involves huge alphabets. In this note we give a short cyclic morphism…

We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite…