Related papers: On the number of collinear triples in permutations

Let $\alpha, \beta \geq 0$ and $\alpha + \beta < 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^\beta(p_{n+1}^\alpha - p_n^\alpha) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of S\'{a}ndor…

The set of all permutations with $n$ symbols is a symmetric group denoted by $S_n$. A transposition tree, $T$, is a spanning tree over its $n$ vertices $V_T=${$1, 2, 3, \ldots n$} where the vertices are the positions of a permutation $\pi$…

Let $\mathcal{F}_n(X;G)$ denote the set of number fields of degree $n$ with absolute discriminant no larger than $X$ and Galois group $G$. This set is known to be finite for any finite permutation group $G$ and $X \geq 1$. In this paper, we…

We find explicit bounds on the primary components and on the Castelnuovo-Mumford regularity of powers of monomial ideals. We also analyze the primary decompositions of Katzman's example.

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')<k$ ($\chi_\ell(G')< k$, $\chi_\mathrm{DP}(G')<k$). Let $f(n,…

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm L}_n(q)$ is prime. We present…

A binary matrix is a matrix with entries from the set $\{0,1\}$. We say that a binary matrix $A$ contains a binary matrix $S$ if $S$ can be obtained from $A$ by removal of some rows, some columns, and changing some $1$-entries to…

We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +\alpha X^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$,…

The collision problem is to decide whether a function X:{1,..,n}->{1,..,n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Theta(n^{1/5}) on the number of queries needed by a quantum computer to…

This paper contributes to the solution of the Poincare problem, which is to bound the degree of a (generalized algebraic) leaf of a (singular algebraic) foliation of the complex projective plane. The first theorem gives a new sort of bound,…

We obtain bounds on the number of triples that determine a given pair of dot products arising in a vector space over a finite field or a module over the set of integers modulo a power of a prime. More precisely, given $E\subset \mathbb…

A permutation graph is a graph that can be derived from a permutation, where the vertices correspond to letters of the permutation, and the edges represent inversions. We provide a construction to show that there are infinitely many…

It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large…

Let $x$ and $n$ be positive integers. We prove a non-trivial lower bound for $x$, dependant only on $\omega_n$, the number of distinct prime factors of $x^n-1$. By considering the divisibility of $\varphi \mid x^n-1$ for $\varphi \mid n$,…

Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = \alpha^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this…

We prove a "decomposition lemma" that allows us to count preimages of certain sets of permutations under West's stack-sorting map $s$. As a first application, we give a new proof of Zeilberger's formula for the number of 2-stack-sortable…

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on…

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…

We study the differential uniformity of the Wan-Lidl polynomials over finite fields. A general upper bound, independent of the order of the field, is established. Additional bounds are established in settings where one of the parameters is…