Related papers: On the Cycle Structure of the Metacommutation Map

We present a general theorem characterizing the relationship between the prime base $p$ representations of non-negative integers $x$ and their positive integer powers, $x^n$. For any positive integer $l$, the theorem establishes the…

Let $X_1,..., X_n$ be independent, uniformly random points from $[0,1]^2$. We prove that if we add edges between these points one by one by order of increasing edge length then, with probability tending to 1 as the number of points $n$…

Let $G$ be a finite group. We show that if $|G| = pqrs$, where $p$, $q$, $r$, and $s$ are distinct odd primes, then every connected Cayley graph on $G$ has a hamiltonian cycle.

We derive several existence results concerning cycle types and, more generally, the "mapping behavior" of complete mappings. Our focus is on so-called first-order cyclotomic mappings, which are functions on a finite field $\mathbb{F}_q$…

Informed by our understanding of the tt-geometry of permutation modules, we investigate the proper definition of the `stable permutation category' of a finite group. Then we prove that this category decomposes over cyclic and generalized…

Berggren and Popovi\'c introduced quadratic-permutation-polynomial interleaved Zadoff--Chu sequences and, from exhaustive data, conjectured that all normalized QPP-interleaved Zadoff--Chu sequences are inequivalent to ordinary Zadoff--Chu…

We study the combinatorial types of periodic orbits of the standard covering endomorphisms ${\mathbf m}_k(x)=k x \ (\text{mod} \ {\mathbb Z})$ of the circle for integers $k \geq 2$ and the frequency with which they occur. For any $q$-cycle…

A cyclic permutation $\pi:\{1, \dots, N\}\to \{1, \dots, N\}$ has a \emph{block structure} if there is a partition of $\{1, \dots, N\}$ into $k\notin\{1,N\}$ segments (\emph{blocks}) permuted by $\pi$; call $k$ the \emph{period} of this…

Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$…

Permutation $\sigma$ appears in permutation $\pi$ if there exists a subsequence of $\pi$ that is order-isomorphic to $\sigma$. The natural question is to check if $\sigma$ appears in $\pi$, and if so count the number of occurrences. We know…

It was shown by Kutnar and \v Sparl in 2009 that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. In this paper, it will be shown that every such graph contains a Hamilton cycle, except…

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central…

For a graph with edge ordering, a linear order on the edge set, we obtain a permutation of vertices by considering the edges as transpositions of endvertices. It is known from D\'enes' results that the permutation of a tree is a full cyclic…

We introduce a new class of transitive permutation groups which properly contains the automorphism groups of vertex-transitive graphs and digraphs. We then give a sufficient condition for a quotient of this family to remain in the family,…

Let p_n denote the sequence of all primes and let d_n=p_n-p_{n-1} denote the sequence of all gaps between consecutive primes. In 1948 Erd\H{o}s and Tur\'an showed that d_{n+1}-d_n changes sign infinitely often and together with P\'olya…

It is a classical result that a random permutation of $n$ elements has, on average, about $\log n$ cycles. We generalise this fact to all directed $d$-regular graphs on $n$ vertices by showing that, on average, a random cycle-factor of such…

We study higher-dimensional homological analogues of bond percolation on a square lattice and site percolation on a triangular lattice. By taking a quotient of certain infinite cell complexes by growing sublattices, we obtain finite cell…

In the present paper, a new type of mappings called perimetric contractions on $k$-polygons is introduced. These contractions can be viewed as a generalization of mappings that contracts perimeters of triangles. A fixed point theorem for…

Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant…

We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the…