On the frequency of permutations containing a long cycle (original) (raw)

Permutations with Restricted Cycle Structure and an Algorithmic Application

Combinatorics, Probability and Computing, 2002

Let q be an integer with q [ges ] 2. We give a new proof of a result of Erdös and Turán determining the proportion of elements of the finite symmetric group Sn having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group An. In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.We apply these results to estimate the proportion of elements of order 2f in Sn, and of order 3f in An and Sn, where gcd(2, f) = 1, and gcd(3, f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the fth power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in An or Sn, given as a black-box group with an order oracle, is discussed.

Identifying long cycles in finite alternating and symmetric groups acting on subsets

Let HHH be a permutation group on a set ΛΛΛ, which is permutationally isomorphic to a finite alternating or symmetric group AnA_nAn or SnS_nSn acting on the k−k-k−element subsets of points from 1,...,n{1,. .. , n}1,...,n, for some arbitrary but fixed kkk. Suppose moreover that no isomorphism with this action is known. We show that key elements of HHH needed to construct such an isomorphism ϕϕϕ, such as those whose image under ϕϕϕ is an n−n-n−cycle or (n−1)−(n − 1)-(n−1)−cycle, can be recognised with high probability by the lengths of just four of their cycles in ΛΛΛ.

Local probabilities for random permutations without long cycles

We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the saddle point method combined with ideas originated in analytic number theory. Theorem 1 and its detailed proof are included to rectify formulas for small r which have been announced by a few other authors.

A black-box group algorithm for recognizing finite symmetric and alternating groups, I

Transactions of the American Mathematical Society, 2003

We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree n of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of n is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of Sn: the conditional probability that a random element σ ∈ Sn is an n-cycle, given that σ n = 1, is at least 1/10.

THE NUMBER OF k-CYCLES IN A FAMILY OF RESTRICTED PERMUTATIONS

In this paper we study different restrictions imposed over the set of permutations of size n, Sn , and for specific classes of restrictions study the cycle structure of corresponding permutations. More specifically, we prove that for any fixed positive integer k, the number of k-cycles of a uniformly chosen permutation π ∈ Sn with the restriction " π(i) ≥ i − 1 for i ∈ {2, ..., n} " has a Normal asymptotic distribution. We further prove that this result translates into CLTs regarding multiplicities of fixed-size parts of a uniformly selected composition of n.

Bounds for the probability of generating the symmetric and alternating groups

Archiv der Mathematik, 2011

We give explicit, asymptotically sharp bounds for the probability that a pair of random permutations of degree n generates either Sn or An and also for the probability that a pair of random even permutations of degree n generates An. As an application we answer a question of Wiegold in the case of alternating groups.

The random (n-k)-cycle to transpositions walk on the symmetric group

arXiv (Cornell University), 2017

We study the rate of convergence of the Markov chain on Sn which starts with a random (n−k)-cycle for a fixed k ≥ 1, followed by random transpositions. The convergence to the stationary distribution turns out to be of order n. We show that after cn + ln k 2 n steps for c > 0, the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the (n − 1)-cycle case. The upper bound relies on estimates for the difference of normalized characters.

Probabilities of permutation equalities in finite groups

Let G be a …nite group and be a permutation from Sn. We investigate the distribution of the probabilities of the equality a1a2 an 1an = a 1 a 2 a n 1 a n when varies over all the permutations in Sn. The probability P r (G) = P r(a1a2 an 1an = a 1 a 2 a n 1 a n) is identical to P r ! 1 (G), with ! = a1a2:::an 1ana 1 1 a 1 2 a 1 n 1 a 1 n ; as it is de…ned in [6] and [19]. The notion of commutativity degree, or the probability of a permutation equality a1a2 = a2a1, for which n = 2 and = h2 1i, was introduced and assessed by P. Erdös and P. Turan in [11] in 1968 and by W. H. Gustafson in [13] in 1973. In [13] Gustafson establishes a relation between the probability of a1; a2 2 G commuting and the number of conjugacy classes in G. In this work we de…ne several other parameters, which depend only on a certain interplay between the conjugacy classes of G, and compute the probabilities of general permutation equalities in terms of these parameters. It turns out that this probability, for a permutation , depends only on the number c(Gr()) of the alternating cycles in the cycle graph Gr() of. The cycle graph of a permutation was introduced by V. Bafna and P. A. Pevzner in [1]. We describe the spectrum of the probabilities of permutation equalities in a …nite group as varies over all the elements of Sn. This spectrum turns-out to be closely related to the partition of n! into a sum of the corresponding Hultman numbers.

The Random (n-k)$$ ( n - k ) -Cycle to Transpositions Walk on the Symmetric Group

Journal of Theoretical Probability, 2018

We study the rate of convergence of the Markov chain on S n which starts with a random (n − k)-cycle for a fixed k ≥ 1, followed by random transpositions. The convergence to the stationary distribution turns out to be of order n. We show that after cn + ln k 2 n steps for c > 0, the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the (n − 1)-cycle case. The upper bound relies on estimates for the difference of normalized characters.