Exact upper bound for sorting Rn with LE

Abstract

A permutation over alphabet [Formula: see text] is a sequence over [Formula: see text], where every element occurs exactly once. [Formula: see text] denotes symmetric group defined over [Formula: see text]. [Formula: see text] denotes the Identity permutation. [Formula: see text] is the reverse permutation i.e., [Formula: see text]. An operation, that we call as an LE operation, has been defined which consists of exactly two generators: set-rotate that we call Rotate and pair-exchange that we call Exchange (OEIS). At least two generators are the required to generate [Formula: see text]. Rotate rotates all elements to the left (moves leftmost element to the right end) and Exchange is the pair-wise exchange of the two leftmost elements. The optimum number of moves for transforming [Formula: see text] into [Formula: see text] with LE operation are known for [Formula: see text]; as listed in OEIS with identity A048200. However, no general upper bound is known. The contributions of this article are: (a) a novel upper bound for the number of moves required to sort [Formula: see text] with LE has been derived; (b) the optimum number of moves to sort the next larger [Formula: see text] i.e., [Formula: see text] has been computed; (c) an algorithm conjectured to compute the optimum number of moves to sort a given [Formula: see text] has been designed.