Permutations with Restricted Cycle Structure and an Algorithmic Application (original) (raw)

On the frequency of permutations containing a long cycle

Journal of Algebra, 2006

A general explicit upper bound is obtained for the proportion P (n, m) of elements of order dividing m, where n − 1 ≤ m ≤ cn for some constant c, in the finite symmetric group S n. This is used to find lower bounds for the conditional probabilities that an element of S n or A n contains an r-cycle, given that it satisfies an equation of the form x rs = 1 where s ≤ 3. For example, the conditional probability that an element x is an n-cycle, given that x n = 1, is always greater than 2/7, and is greater than 1/2 if n does not divide 24. Our results improve estimates of these conditional probabilities in earlier work of the authors with Beals, Leedham-Green and Seress, and have applications for analysing black-box recognition algorithms for the finite symmetric and alternating groups.

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.

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.

On the proportion of elements of prime order in finite symmetric groups

arXiv (Cornell University), 2022

Dedicated to Daniela Nikolva on the occasion of her 70 th birthday. We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order p, acting on a set of given size n, which is sharp for certain n and p. Namely, we prove that if n ≡ k (mod p) with 0 ≤ k ≤ p − 1, then this proportion is at most (p • k!) −1 with equality if and only if p ≤ n < 2n.

A cycle lemma for permutation inversions

Discrete Mathematics, 2002

In this paper we study some properties of the inversion statistic. Some enumerative results concerning the permutations of the multiset {x m 1 1 ; x m 2 2 } with respect to the inversion parameter are established and it is shown that these depend on gcd(m1; m2). Using a "cycle lemma", a combinatorial proof of the results is given. Moreover, some applications to the Gaussian binomial coe cient are illustrated.

Using Recurrence Relations to Count Certain Elements in Symmetric Groups

European Journal of Combinatorics, 2001

We use the fact that certain cosets of the stabilizer of points are pairwise conjugate in a symmetric group S n in order to construct recurrence relations for enumerating certain subsets of S n. Occasionally one can find 'closed form' solutions to such recurrence relations. For example, the probability that a random element of S n has no cycle of length divisible by q is n/q d=1 (1 − 1 dq).

Most permutations power to a cycle of small prime length

Proceedings of the Edinburgh Mathematical Society

We prove that most permutations of degree nnn have some power which is a cycle of prime length approximately logn\log nlogn. Explicitly, we show that for nnn sufficiently large, the proportion of such elements is at least 1−5/loglogn1-5/\log \log n1−5/loglogn with the prime between logn\log nlogn and (logn)loglogn(\log n)^{\log \log n}(logn)loglogn. The proportion of even permutations with this property is at least 1−7/loglogn1-7/\log \log n1−7/loglogn.

Recursions for excedance number in some permutations groups

The excedance number for S n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive proof which seems to be folklore and extend it to the colored permutation groups G r,n .