Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRedei permutations with cycles of the same length
64
0
0.0
(
0
)
Added by
Juliane Golubinski Capaverde
Publication date
2020
fields
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Let $mathbb{F}_q$ be a finite field of odd characteristic. We study Redei functions that induce permutations over $mathbb{P}^1(mathbb{F}_q)$ whose cycle decomposition contains only cycles of length $1$ and $j$, for an integer $jgeq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a Redei permutation of this type to exist over $mathbb{P}^1(mathbb{F}_q)$, characterize Redei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for Redei involutions based on the number of fixed points, and procedures to construct Redei permutations with a prescribed number of fixed points and $j$-cycles for $j in {3,4,5}$.
rate research
Read More
We compute the limiting distribution, as n approaches infinity, of the number of cycles of length between gamma n and delta n in a permutation of [n] chosen uniformly at random, for constants gamma, delta such that 1/(k+1) <= gamma < delta <= 1/k for some integer k. This distribution is supported on {0, 1, ... k} and has 0th, 1st, ..., kth moments equal to those of a Poisson distribution with parameter log (delta/gamma). For more general choices of gamma, delta we show that such a limiting distribution exists, which can be given explicitly in terms of certain integrals over intersections of hypercubes with half-spaces; these integrals are analytically intractable but a recurrence specifying them can be given. The results herein provide a basis of comparison for similar statistics on restricted classes of permutations.
Recursive permutations whose cycles are the classes of a decidable equivalence relation are studied; the set of these permutations is called $mathrm{Perm}$, the group of all recursive permutations $mathcal{G}$. Multiple equivalent computable representations of decidable equivalence relations are provided. $mathcal{G}$-conjugacy in $mathrm{Perm}$ is characterised by computable isomorphy of cycle equivalence relations. This result parallels the equivalence of cycle type equality and conjugacy in the full symmetric group of the natural numbers. Conditions are presented for a permutation $f in mathcal{G}$ to be in $mathrm{Perm}$ and for a decidable equivalence relation to appear as the cycle relation of a member of $mathcal{G}$. In particular, two normal forms for the cycle structure of permutations are defined and it is shown that conjugacy to a permutation in the first normal form is equivalent to membership in $mathrm{Perm}$. $mathrm{Perm}$ is further characterised as the set of maximal permutations in a family of preordered subsets of automorphism groups of decidable equivalences. Conjugacy to a permutation in the second normal form corresponds to decidable cycles plus decidable cycle finiteness problem. Cycle decidability and cycle finiteness are both shown to have the maximal one-one degree of the Halting Problem. Cycle finiteness is used to prove that conjugacy in $mathrm{Perm}$ cannot be decided and that it is impossible to compute cycle deciders for products of members of $mathrm{Perm}$ and finitary permutations. It is also shown that $mathrm{Perm}$ is not recursively enumerable and that it is not a group.
In this article, we discuss whether a single congruent number $t$ can have two (or more) distinct triangles with the same hypotenuse. We also describe and carry out computational experimentation providing evidence that this does not occur.
Let $pequiv1pmod 4$ be a prime. In this paper, with the help of Jacobsthal sums, we study some permutation problems involving biquadratic residues modulo $p$.
We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let $c(ell)$ be the limit of the ratio of the maximum number of cycles of length $ell$ in an $n$-vertex tournament and the expected number of cycles of length $ell$ in the random $n$-vertex tournament, when $n$ tends to infinity. It is well-known that $c(3)=1$ and $c(4)=4/3$. We show that $c(ell)=1$ if and only if $ell$ is not divisible by four, which settles a conjecture of Bartley and Day. If $ell$ is divisible by four, we show that $1+2cdotleft(2/piright)^{ell}le c(ell)le 1+left(2/pi+o(1)right)^{ell}$ and determine the value $c(ell)$ exactly for $ell = 8$. We also give a full description of the asymptotic structure of tournaments with the maximum number of cycles of length $ell$ when $ell$ is not divisible by four or $ellin{4,8}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
