Related papers: New Lower Bounds for Permutation Arrays Using Cont…

We study $P(n,3)$, the size of the largest subset of the set of all permutations $S_n$ with minimum Kendall $\tau$-distance $3$. Using a combination of group theory and integer programming, we reduced the upper bound of $P(p,3)$ from…

A set $S$ of permutations is forcing if for any sequence $\{\Pi_i\}_{i \in \mathbb{N}}$ of permutations where the density $d(\pi,\Pi_i)$ converges to $\frac{1}{|\pi|!}$ for every permutation $\pi \in S$, it holds that $\{\Pi_i\}_{i \in…

Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$,…

An (n,d)-permutation code is a subset C of Sym(n) such that the Hamming distance d_H between any two distinct elements of C is at least equal to d. In this paper, we use the characterisation of the isometry group of the metric space…

A frequency permutation array (FPA) of length $n=m\lambda$ and distance $d$ is a set of permutations on a multiset over $m$ symbols, where each symbol appears exactly $\lambda$ times and the distance between any two elements in the array is…

Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the…

We consider rational functions of the form $V(x)/U(x)$, where both $V(x)$ and $U(x)$ are polynomials over the finite field $\mathbb{F}_q$. Polynomials that permute the elements of a field, called {\it permutation polynomials ($PPs$)}, have…

The Cayley distance between two permutations $\pi, \sigma \in S_n$ is the minimum number of \textit{transpositions} required to obtain the permutation $\sigma$ from $\pi$. When we only allow adjacent transpositions, the minimum number of…

A number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with the connected problems of sorting permutations in "as few moves as possible", using a given set of allowed operations, or…

In this article we answer a question asked by Chien et al. in arXiv:2304.06050 in which they study the numerical range of weighted cyclic matrices under permutation of their entries. Namely, we are interested in how $w(A_\sigma)$ fluctuates…

Let $S_n$ be the symmetric group on the set $[n]:=\{1,2,\ldots,n\}$. Given a permutation $\sigma=\sigma_1\sigma_2 \cdots \sigma_n \in S_n$, we say it has a descent at index $i$ if $\sigma_i>\sigma_{i+1}$. Let $\mathcal{D}(\sigma)$ be the…

Given positive integers $n$ and $d$, let $M(n,d)$ denote the maximum size of a permutation code of length $n$ and minimum Hamming distance $d$. The Gilbert-Varshamov bound asserts that $M(n,d) \geq n!/V(n,d-1)$ where $V(n,d)$ is the volume…

We introduce twisted permutation codes, which are frequency permutation arrays analogous to repetition permutation codes, namely, codes obtained from the repetition construction applied to a permutation code. In particular, we show that a…

Permutation arrays under the Chebyshev metric have been considered for error correction in noisy channels. Let $P(n,d)$ denote the maximum size of any array of permutations on $n$ symbols with pairwise Chebyshev distance $d$. We give new…

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…

The sequence reconstruction problem asks for the recovery of a sequence from multiple noisy copies, where each copy may contain up to $r$ errors. In the case of permutations on \(n\) letters under the Hamming metric, this problem is closely…

A permutation $\sigma\in S_n$ is said to be $k$-universal or a $k$-superpattern if for every $\pi\in S_k$, there is a subsequence of $\sigma$ that is order-isomorphic to $\pi$. A simple counting argument shows that $\sigma$ can be a…

Permutation codes are widely studied objects due to their numerous applications in various areas, such as power line communications, block ciphers, and the rank modulation scheme for flash memories. Several kinds of metrics are considered…

Permutons are probability measures on the unit square with uniform marginals that provide a natural way to describe limits of permutations. We are interested in the permuton limits for permutations sampled uniformly from certain…

In order to overcome the challenges caused by flash memories and also to protect against errors related to reading information stored in DNA molecules in the shotgun sequencing method, the rank modulation is proposed. In the rank modulation…