Related papers: Bicrucial $k$-power-free permutations

Two words have a reverse if they have the same pair of distinct letters on the same pair of positions, but in reversed order. A set of words no two of which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size of a…

In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[\alpha n]$, $[\alpha n]+1$, where $\alpha>1$ is irrational number with bounded partial quotient or irrational algebraic number.

Let $G$ be a non-trivial torsion free group and $t$ be an unknown. In this paper we consider three equations (over $G$) of arbitrary length and show that they have a solution (over $G$) provided two relations among their coefficients hold.…

We compute the number of ways a given permutation can be written as a product of exactly $k$ transpositions. We express this number as a linear combination of explicit geometric sequences, with coefficients which can be computed in many…

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive…

In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form $[\alpha p]$, $[\alpha p]+1$, where $p$ is prime and $\alpha>0$ is irrational algebraic number. We also establish an asymptotic…

It is shown that the maximum number of patterns that can occur in a permutation of length $n$ is asymptotically $2^n$. This significantly improves a previous result of Coleman.

Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for…

Recently, B\'ona and Smith defined strong pattern avoidance, saying that a permutation $\pi$ strongly avoids a pattern $\tau$ if $\pi$ and $\pi^2$ both avoid $\tau$. They conjectured that for every positive integer $k$, there is a…

In this note, we prove by using T. Estermann's and S. Dimitrov's arguments with an elementary inequality that there are infinitely many $n$ for which all of the numbers $n^2+1,n^2+2$ and $n^2+3$ are squarefree. We also improve the error…

By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii)…

We prove effective finiteness results concerning polynomial values of the sums $$ b^k +\left(a+b\right)^k + \cdots + \left(a\left(x-1\right) + b\right)^k $$ and $$ b^k - \left(a+b\right)^k + \left(2a+b\right)^k - \ldots + (-1)^{x-1}…

A \emph{power} is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer; the power is also called a {\em $k$-power} and $k$ is its {\em exponent}. We prove that for any $k \ge 2$,…

We provide a combinatorial approach to the largest power of $p$ in the number of permutations $\pi$ with $\pi^p=1$, for a fixed prime number $p$. With this approach, we find the largest power of $2$ in the number of involutions, in the…

We introduce an elementary congruence-based procedure to look for q-th power multiples in arbitrary binary recurrence sequences (q>2). The procedure allows to prove that no such multiples exist in many instances.

We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a…

Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the…

We analyze the structure and enumerate Dumont permutations of the first and second kinds avoiding certain patterns or sets of patterns of length 3 and 4. Some cardinalities are given by Catalan numbers, powers of 2, little Schroeder…

We develop a general framework for finding all perfect powers in sequences derived by shifting non-degenerate quadratic Lucas-Lehmer binary recurrence sequences by a fixed integer. By combining this setup with bounds for linear forms in…

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…