We study sorting operators A on permutations that are obtained composing Knuth’s stack sorting operator S and the reversal operator R, as many times as desired. For any such operator A, we provide a size-preserving bijection between the set of permutations sorted by S ◦ A and the set of those sorted by S ◦ R ◦ A, proving that these sets are enumerated by the same sequence, but also that many cl...

Permutations are a core component of almost every cipher. No matter if we consider the DES [5], AES [2] or most of the other encryption algorithms [6] relevant nowadays, we always find permutation operators as essential building blocks inside. In this contribution we will introduce key-dependent permutation operators of provably excellent quality inspired by chaotic Kolmogorov flows [1, 4]. Fro...

We introduce the Dual Bubble Sort operator B̂ (a sorting algorithm such that, if σ = α 1β is a permutation, then B̂(σ) = 1α B̂(β)) and consider the set of permutations sorted by the composition B̂B, where B is the classical Bubble Sort operator. We show that this set is a permutation class and we determine the generating function of the descent and fixed point distributions over this class. Afterwa...

In this paper we examine the sorting operator T (LnR) = T (R)T (L)n. Applying this operator to a permutation is equivalent to passing the permutation reversed through a stack. We prove theorems that characterise t-revstack sortability in terms of patterns in a permutation that we call zigzag patterns. Using these theorems we characterise those permutations of length n which are sorted by t appl...

In order to sequence the tasks of a job shop problem (JSP) on a number of machines related to the technological machine order of jobs, a new representation technique { mathematically known as \permutation with repetition" is presented. The main advantage of this single chromosome representation is { in analogy to the permutation scheme of the traveling salesman problem (TSP) { that it cannot pr...

This paper presents a new evolutionary system using genetic algorithm for evolving artificial neural networks (ANNs). The proposed algorithm is “Permutation free Encoding Technique for Evolving Neural Networks”(PETENN) that uses a novel encoding scheme for representing ANNs. Existing genetic algorithms (GAs) for evolving ANNs suffer from the permutation problem, resulting from the recombination...