Related papers: Characterization of quasirandom permutations by a …

Let $S_{\rm lcm}(n)$ denote the set of permutations $\pi$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,\pi(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $\pi$ of $[n]$ such that…

We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced…

We show that for every sufficiently large $n$, the number of monotone subsequences of length four in a permutation on $n$ points is at least $\binom{\lfloor n/3 \rfloor}{4} + \binom{\lfloor(n+1)/3\rfloor}{4} + \binom{\lfloor…

How many permutations are needed so that every infinite-coinfinite set of natural numbers with asymptotic density can be rearranged to no longer have the same density? We prove that the density number $\mathfrak{dd}$, which answers this…

In this paper, we introduce a convergence notion for ordered selections. Our convergence notion is based on subpermutation densities and convergences of the marginal distributions. A particular case of this convergence is the well-known…

For every fixed graph $H$ and every fixed $0 < \alpha < 1$, we show that if a graph $G$ has the property that all subsets of size $\alpha n$ contain the ``correct'' number of copies of $H$ one would expect to find in the random graph…

We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for…

A permutation $\pi$ contains a permutation $\sigma$ as a pattern if it contains a subsequence of length $|\sigma|$ whose elements are in the same relative order as in the permutation $\sigma$. This notion plays a major role in enumerative…

We say that permutations $\pi_1,\dots, \pi_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $\pi'_1,\dots,\pi'_r$ of these permutations, $\pi'_1,\dots,\pi'_r$ generate $\mathcal{S}_n$. We…

We show that the number of members of S_n avoiding any one of five specific triples of 4-letter patterns is given by sequence A111279 in OEIS, which is known to count weak sorting permutations. By numerical evidence, there are no other…

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and…

There is a deep connection between permutations and trees. Certain sub-structures of permutations, called sub-permutations, bijectively map to sub-trees of binary increasing trees. This opens a powerful tool set to study enumerative and…

The relation of s-convexity and sets modeling physical quasicrystals is explained for quasicrystals related to quadratic unitary Pisot numbers. We show that 1-dimensional model sets may be characterized by s-convexity for finite set of…

For a set of permutations $S\subseteq S_n$, consider the quasisymmetric generating function $$Q(S): = \sum_{w\in S}F_{n, \mathrm{Des}(w)},$$ where $\mathrm{Des}(w) := \{i\mid w(i)> w(i+1)\}$ is the descent set of $w$ and $F_{n,…

A permutation may be represented by a collection of paths in the plane. We consider a natural class of such representations, which we call tangles, in which the paths consist of straight segments at 45 degree angles, and the permutation is…

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our…

The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$. This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by Liu.

Let $\mathcal{P}$ denote the set of all primes, and let $\underline\delta(P)$ denote the relative lower density of a subset $P$ in $\mathcal{P}$. Suppose that $P_1, P_2, P_3, P_4$ are four subsets of primes with…

For a given shape $S$ in the plane, one can ask what is the lowest possible density of a point set $P$ that pierces ("intersects", "hits") all translates of $S$. This is equivalent to determining the covering density of $S$ and as such is…

Let $\mathbb{K}$ be a field of characteristic $0$. For each choice of distinct $a_1, \ldots, a_n\in \mathbb{K}$ and distinct $b_1, \ldots, b_n\in \mathbb{K}$, consider the sum $S=\sum_{i=1}^n a_i b_{\pi(i)}$ as $\pi$ ranges over the…