Related papers: On the number of collinear triples in permutations

We consider the set $\mathcal{M}_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic…

We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $\mathbb R^2$, with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main…

Let $n$ be a positive integer and let $[n]=\{1,2,\ldots,n\}$. Let $\Gamma_n$ denote the group of permutations on $[n]$ whose restrictions to maximal proper subsets of $[n]$ are even, let $\Sigma_n$ denote the monoid of transformations on…

Let n>=3 be an odd integer. For any integer a prime to n, define the permutation gamma_{a,n} of {1,...,(n-1)/2} by gamma_{a,n}(x)=n-\dec{ax}_n if {ax}_n>=(n+1)/2, and {ax}_n if {ax}_n<=(n-1)/2, where {x}_n denotes the least nonnegative…

Let $\alpha(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $\alpha(2n+1) = (2n+1) \alpha(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents…

We prove an upper bound for the number of cyclic transitive subgroups in a finite permutation group and clarify the structure of the groups for which this bound becomes sharp. We also give an application in the theory of number fields.

Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it…

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n} \{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with…

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael…

Two results (together with their relatively elementary proofs) are presented. The first one presents the upper boundary on the number of spanning trees in a finite planar multigraph, proving that the complexity (the number of spanning…

Let $k\geq1$ be a fixed integer, and $\mathcal P_N$ be the set of primes no more than $N$. We prove that if a set $\mathcal A\subset\mathcal P_N$ contains no patterns $p_1,p_1+(p_2-1)^k$, where $p_1,p_2$ are prime numbers, then \[…

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has…

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits,…

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq…

It is well known that $n/(n - \mu)$, where $\mu$ is the spectral radius of a graph with $n$ vertices, is a lower bound for the clique number. We conjecture that $\mu$ can be replaced in this bound with $\sqrt{s^+}$, where $s^+$ is the sum…

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.

For various quadruple systems F, we give asymptotically sharp lower bounds on the number of copies of F in a quadruple system with a prescribed number of vertices and edges. Our results extend those of Furedi, Keevash, Pikhurko, Simonovits…

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log\log n$ with…

We find a lower bound for the number of Chen primes in the arithmetic progression $a \bmod q$, where $(a,q)=(a+2,q)=1$. Our estimate is uniform for $q \leq \log^M x$, where $M>0$ is fixed.

This article studies the poset of simple permutations with respect to the pattern involvement. We specify results on critically indecomposable posets obtained by Schmerl and Trotter to simple permutations and prove that if $\sigma, \pi$ are…