Related papers: Discriminators and k-Regular Sequences

Let $n$ be a positive integer and $I$ a $k$-subset of integers in $[0,n-1]$. Given a $k$-tuple $A=(\alpha_0, \cdots, \alpha_{k-1})\in \mathbb{F}^k_{q^n}$, let $M_{A,I}$ denote the matrix $(\alpha_i^{q^j})$ with $0\leq i\leq k-1$ and $j\in…

We study the primitive divisors of the terms of $(\Delta_n)_{n \geq 1}$, where $\Delta_n=N_{K/ \mathbb{Q}}(u^n-1)$ for $K$ a real quadratic field, and $u>1$ a unit element of its ring of integers. The methods used allow us to find the terms…

A homogeneous polynomial S(x_1, ..., x_n) of degree r in n variables posesses a discriminant D_{n|r}(S), which vanishes if and only if the system of equations dS/dx_i = 0 has non-trivial solutions. We give an explicit formula for…

Inversion sequences are integer sequences $(\sigma_1, \dots, \sigma_n)$ such that $0 \leqslant \sigma_i < i$ for all $1 \leqslant i \leqslant n$. The study of pattern-avoiding inversion sequences began in two independent articles by…

Let $d_1 = 1 < d_2 < d_3 < \cdots < d_{\tau(n)} = n$ denote the increasing sequence of the divisors of a positive integer $n$. In this paper, for real or complex values of $\alpha$, we define and study some properties of two new divisor…

Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has…

Let $n$ and $k$ be positive integers, and let $F$ be an alphabet of size $n$. A sequence over $F$ of length $m$ is a \emph{$k$-radius sequence} if any two distinct elements of $F$ occur within distance $k$ of each other somewhere in the…

We consider two infinite classes of ordinary difference equations admitting Lax pair representation. Discrete equations in these classes are parameterized by two integers $k\geq 0$ and $s\geq k+1$. We describe the first integrals for these…

We generalize the Stirling numbers of the first kind $s(a,k)$ to the case where $a$ may be an arbitrary real number. In particular, we study the case in which $a$ is an integer. There, we discover new combinatorial properties held by the…

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$, leading this author and…

Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An $n\times n$ matrix $A$ is called oscillatory if all its minors are nonnegative and there exists a positive integer $k$ such that all minors of $A^k$ are…

For many types of graphs, criteria have been discovered that give necessary and sufficient conditions for an integer sequence to be the degree sequence of such a graph. These criteria tend to take the form of a set of inequalities, and in…

We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$,…

Schreier sets have been an object of study since first introduced in 1930 by Jozef Schreier to construct a counterexample to a conjecture of Banach. In 1974 George Andrews found interesting connections between these sets and Fibonacci…

For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic sequence…

For integers m >= 1, s >= 0, and t >= 1, let K_s + mK_t denote the join of a clique K_s and m vertex-disjoint copies of K_t. We prove that for fixed m >= 1, t >= 1, and s >= 0, R(K_s + mK_t, K_n) = O( n^{s+t-1} / (log n)^{s+t-2} ). This…

Let $\Sigma$ be a countable alphabet. For $r\geq 1$, an infinite sequence $s$ with characters from $\Sigma$ is called $r$-quasi-regular, if for each $\sigma\in\Sigma$ the ratio of the longest to shortest interval between consecutive…

In this note we give a brief survey of the most elementary criteria used to determine the surjectivity of the trace operator on the ring of integers of a number field $K$. Furthermore, we introduce an easy to state yet unknown surjectivity…

Let $N$ be any fixed positive integer and define \begin{align*} S_N(x)=\sum_{m, n \leq x}d(n^2+Nm^2), \end{align*} where $d(n)$ is the divisor function. We evaluate asymptotically $S_N(x)$ for several $N$, extending earlier works of Gafurov…

A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and two…