The set of cycle lengths of almost all permutations in Sn are “Poisson distributed”: we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain “normal order” (in the spirit of the Erdős-Turán theorem). Our results were inspired by analogous ...

Recently, Kalikinkar Mandal and Guang Gong presented a family of nonlinear pseudorandom number generators using Welch-Gong Transformations in their paper [6]. They also performed the cycle decomposition of the WG-NLFSR recurrence relations over different finite fields by computer simulations where the nonlinear recurrence relation is composed of a characteristic polynomial and a WG permutation....

Berry Esseen type bounds to the normal, based on zeroand size-bias couplings, are derived using Stein’s method. The zero biasing bounds are illustrated with an application to combinatorial central limit theorems where the random permutation has either the uniform distribution or one which is constant over permutations with the same cycle type and having no fixed points. The size biasing bounds ...

We give the first Gray code for the set of n-length permutations with a given number of cycles. In this code, each permutation is transformed into its successor by a product with a cycle of length three, which is optimal. If we represent each permutation by its transposition array then the obtained list still remains a Gray code and this allows us to construct a constant amortized time (CAT) al...

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n=p for an odd prime p. We construct a family of (p−1)/2 non-isomorphic perfect 1-factorisations of Kn, n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltonian if the permutation defined by any row relative to any other row is a single cy...

Berry Esseen type bounds to the normal, based on zeroand size-bias couplings, are derived using Stein’s method. The zero biasing bounds are illustrated with an application to combinatorial central limit theorems where the random permutation has either the uniform distribution or one which is constant over permutations with the same cycle type and having no fixed points. The size biasing bounds ...

An involution is a permutation that is its own inverse. Given a permutation σ of [n], let Nn(σ) denote the number of ways to write σ as a product of two involutions of [n]. If we endow the symmetric groups Sn with uniform probability measures, then the random variables Nn are asymptotically lognormal. The proof is based upon the observation that, for most permutations σ, Nn(σ) can be well appro...

—An algorithm for finding the starting state of a feedback tailbiting encoder is proposed, based solely on the properties of the encoder state diagram. With this approach, conditions for the existence of a tailbiting code generated by such an encoder are naturally tested in the process of preparing a certain permutation table for the starting state determination algorithm. A tailbiting failure ...

Berry–Esseen-type bounds to the normal, based on zeroand size-bias couplings, are derived using Stein’s method. The zero biasing bounds are illustrated in an application to combinatorial central limit theorems in which the random permutation has either the uniform distribution or one that is constant over permutations with the same cycle type, with no fixed points. The size biasing bounds are a...

Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V⊗k . We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V⊗k are isomorphic to the categories of weak, weak admissible and ordinary V -modules, respectively. The main result is an explicit...