Related papers: Multiple contractions of permutation arrays

In this paper, we study some properties of a certain kind of permutation $\sigma$ over $\mathbb{F}_{2}^{n}$, where $n$ is a positive integer. The desired properties for $\sigma$ are: (1) the algebraic degree of each component function is…

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…

Graham-Lov\'asz-Pollak \cite{GL,GP} obtained the celebrated formula $$\det({\sf D}(T_{n+1}))=(-1)^nn2^{n-1},$$ for the determinant of the distance matrix ${\sf D}(T_{n+1})$ for any tree $T_{n+1}$ with $n+1$ vertices. Later, Hou and Woo…

The text-to-pattern Hamming distances problem asks to compute the Hamming distances between a given pattern of length $m$ and all length-$m$ substrings of a given text of length $n\ge m$. We focus on the $k$-mismatch version of the problem,…

We find exact and asymptotic formulas for the number of pairs $(p,q)$ of $N$-cycles such that the all cycles of the product $p\cdot q$ have lengths from a given integer set. We then apply these results to prove a surprisingly high lower…

Consider a random graph $G$ of size $N$ constructed according to a \textit{graphon} $w \, : \, [0,1]^{2} \mapsto [0,1]$ as follows. First embed $N$ vertices $V = \{v_1, v_2, \ldots, v_N\}$ into the interval $[0,1]$, then for each $i < j$…

We consider a random permutation drawn from the set of permutations of length $n$ that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after suitable…

The symmetrization map $\pi:\mathbb C^2\rightarrow \mathbb C^2$ is defined by $ \pi(z_1,z_2)=(z_1+z_2,z_1z_2). $ The closed symmetrized bidisc $\Gamma$ is the symmetrization of the closed unit bidisc $\overline{\mathbb D^2}$, that is, \[…

Let $p$ be a prime number and $\varsigma$ and $m$ be a positive integers. Let $\mathcal{R} = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$ ($u^3 = 0$). Cyclic codes of length $2^\varsigma$ over $\mathcal{R}$ are precisely the…

The shift distance $\mathsf{sh}(S_1,S_2)$ between two strings $S_1$ and $S_2$ of the same length is defined as the minimum Hamming distance between $S_1$ and any rotation (cyclic shift) of $S_2$. We study the problem of sketching the shift…

In this short note we give a new upper bound for the size of a set family with a single Hamming distance. Our proof is an application of the linear algebra bound method.

It is shown that if a non-zero function $f\in B_\sigma$ has infinitely many double zeros on the real axis, then there exists at least one pair of consecutive zeros whose distance apart is greater than $\dfrac{\pi}{\sigma}\tau^{1/4}$,…

A major problem in the study of combinatorial aspects of permutation groups is to determine the distances in the symmetric group $\Sym_n$ with respect to a generator set. One well-known such a case is when the generator set $S_n$ consists…

Let $\mu$ be a doubling measure in $\mathbb{R}^n$. We investigate quantitative relations between the rectifiability of $\mu$ and its distance to flat measures. More precisely, for $x$ in the support $\Sigma$ of $\mu$ and $r > 0$, we…

Cohn and Kumar showed that the permutation on the set of the classes of left associated Hurwitz primes above an odd prime $p$ induced through metacommutation by a Hurwitz prime $\xi$ of norm $q$ has either $0$, $1$ or $2$ fixed points, and…

In this paper, we use Hasse diagrams and generating functions to count alternating permutations with restricted prefix and suffix of lengths 3 and 4. In other words, for an alternating permutation…

We introduce a metric on the set of permutations of given order, which is a weighted generalization of Kendall's $\tau$ rank distance and study its properties. Using the edge graph of a permutohedron, we give a criterion which guarantees…

Given an integer $m\geq1$. Let $\Sigma^{(m)}=\{1,2, \cdots, m\}^{\mathbb{N}}$ be a symbolic space, and let $\{(b_{k},D_{k})\}_{k=1}^{m}:=\{(b_{k}, \{0,1,\cdots, p_{k}-1\}t_{k}) \}_{k=1}^{m}$ be a finite sequence pairs, where integers $|…

Let m be a fixed positive integer. It is well-known that a permutation $\sigma$ may have one, many, or no mth roots. In this note we provide an explicit expression and a generating function for the number of mth roots of \sigma. Let p_m(n)…

We study some analytic properties of distance decreasing self-maps onto the complement of a smooth curve $\Sigma$ in $S^n$. For $n>4$ and $n\equiv 0 \mod 4$, let $\Sigma$ be an embedded circle in $S^n$ and let $g$ be a complete Riemannian…