Related papers: R\'edei permutations with the same cycle structure

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring, and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $R$-linear category $\mathcal{F}^\Delta_{Rpp_k}$ of…

Let $\lambda$ be a partition of an integer $n$ and ${\mathbb F}_q$ be a finite field of order $q$. Let $P_\lambda(q)$ be the number of strictly upper triangular $n\times n$ matrices of the Jordan type $\lambda$. It is known that the…

Let $q=p^n$ be an odd prime power and let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $\widehat{\mathbb{F}_q^{\times}}$ be the group of all multiplicative characters of $\mathbb{F}_q$ and let $\chi$ be a generator of…

Consider a permutation p to be any finite list of distinct positive integers. A statistic is a function St whose domain is all permutations. Let S(p,q) be the set of shuffles of two disjoint permutations p and q. We say that St is shuffle…

We present consistently ordered calculations of the structure functions F_2(x,Q^2) and F_L(x,Q^2), in different expansion schemes. After discussing the standard expansion in powers of alpha_s(Q^2) we consider a leading-order expansion in…

Closed form expressions are proposed for the Feynman integral $$ I_{D, m}(p,q) = \int\frac{d^my}{(2\pi)^m}\int\frac{d^Dx}{(2\pi)^D} \frac1{(x-p/2)^2+(y-q/2)^4} \frac1{(x+p/2)^2+(y+q/2)^4} $$ over $d=D+m$ dimensional space with…

We show that the probability that two permutations of $n$ letters have the same number of cycles is \[\sim \frac{1}{2\sqrt{\pi\log{n}}}.\]

Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor_i^R(M,N) is isomorphic to…

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analyse observed sequences of q-triplets, or q-doublets if one of them is the unity, in terms of cycles of successive M\"obius…

Let $R$ be an associative ring with unity $1$ and consider that $2,k$ and $2k\in \mathbb{N}$ are invertible in $R$. For $m\geq 1$ denote by $UT_n(m,R)$ and $UT_{\infty}(m,R)$, the subgroups of $UT_n(R)$ and $UT_{\infty}(R)$ respectively,…

The resummation of $O(\alpha^{l+1} \ln^{2l} x)$ terms in the evolution kernels of non--singlet combinations of structure functions is investigated for both QED abd QCD. Numerical results are presented for unpolarized and polarized structure…

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension…

Let $\texttt{R}$ be a commutative finite chain ring of invariants $(q,s).$ In this paper, the trace representation of any free cyclic $\texttt{R}$-linear code of length $\ell,$ is presented, via the $q$-cyclotomic cosets modulo $\ell,$ when…

We define and investigate a family of permutations matrices, called shuffling matrices, acting on a set of $N=n_1\cdots n_m$ elements, where $m\geq 2$ and $n_i\geq 2$ for any $i=1,\ldots, m$. These elements are identified with the vertices…

In this paper, we present infinite families of permutations of $\mathbb{F}_{2^{2n}}$ with high nonlinearity and boomerang uniformity $4$ from generalized butterfly structures. Both open and closed butterfly structures are considered. It…

Let $R_q(r,n)$ denote the $r$th order Reed-Muller code of length $q^n$ over $\Bbb F_q$. We consider two algebraic questions about the Reed-Muller code. Let $H_q(r,n)=R_q(r,n)/R_q(r-1,n)$. (1) When $q=2$, it is known that there is a…

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 continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We…

Let $1<t<n$ be integers, where $t$ is a divisor of $n$. An R-$q^t$-partially scattered polynomial is a $\mathbb F_q$-linearized polynomial $f$ in $\mathbb F_{q^n}[X]$ that satisfies the condition that for all $x,y\in\mathbb F_{q^n}^*$ such…

Answering a question of Geoff Robinson, we compute the large n limiting proportion of i(n,q)/q^[n^2/2], where i(n,q) denotes the number of involutions in GL(n,q). We give similar results for the finite unitary, symplectic, and orthogonal…