Related papers: On the constructions of $n$-cycle permutations
We investigate the notion of cyclicity for convolutional codes as it has been introduced by Piret and Roos in the seventies. Codes of this type are described as submodules of the module of all vector polynomials in one variable with some…
Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form…
In this paper, we construct two classes of permutation polynomials over $\mathbb{F}_{q^2}$ with odd characteristic from rational R\'{e}dei functions. A complete characterization of their compositional inverses is also given. These…
Permutation codes are a class of structured vector quantizers with a computationally-simple encoding procedure based on sorting the scalar components. Using a codebook comprising several permutation codes as subcodes preserves the…
In this work we study the acyclic orientations of graphs. We obtain an encoding of the acyclic orientations of the complete $p$-partite graph with size of its parts $n_1,n_2,\ldots,n_p$ via a vector with $p$ symbols and length…
Let $S_n$ be the symmetric group of all permutations of $\{1, \cdots, n\}$ with two generators: the transposition switching $1$ with $2$ and the cyclic permutation sending $k$ to $k+1$ for $1\leq k\leq n-1$ and $n$ to $1$ (denoted by…
Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is…
Let $d\geq 2$ be an integer. In this paper we study arithmetic properties of the sequence $(H_d(n))_{n\in\N}$, where $H_{d}(n)$ is the number of permutations in $S_{n}$ being products of pairwise disjoint cycles of a fixed length $d$. In…
A permutomino of size n is a polyomino determined by a pair of permutations of size n+1, such that they differ in each position. In this paper, after recalling some enumerative results about permutominoes, we give a first algorithm for the…
This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike…
In this paper, we construct some new classes of complete permutation monomials with exponent $d=\frac{q^n-1}{q-1}$ using AGW criterion (a special case). This proves two recent conjectures in [Wuetal2] and extends some of these recent…
We explore various techniques to compress a permutation $\pi$ over n integers, taking advantage of ordered subsequences in $\pi$, while supporting its application $\pi$(i) and the application of its inverse $\pi^{-1}(i)$ in small time. Our…
We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds…
A general and fast method is conceived for computing the cyclic convolution of n points, where n is a prime number. This method fully exploits the internal structure of the cyclic matrix, and hence leads to significant reduction of the…
We study joint distributions of cycles and patterns in permutations written in standard cycle form. We explore both classical and generalised patterns of length 2 and 3. Many extensions of classical theory are achieved; bivariate generating…
A method to construct and count all the linear codes (of arbitrary length) in $\mathbb{F}_{4}$ that are invariant under reverse permutation and that contain the repetition code is presented. These codes are suitable for constructing DNA…
We present a new algorithm for iterating over all permutations of a sequence. The algorithm leverages elementary~$O(1)$ operations on recursive lists. As a result, no new nodes are allocated during the computation. Instead, all elements are…
The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles: \[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $\Omega$.…
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…
We undertake a detailed investigation into the structure of permutations in monotone grid classes whose row-column graphs do not contain components with more than one cycle. Central to this investigation is a new decomposition, called the…