An Upper Bound for Sorting Rn with LE

Abstract

A permutation on a given alphabet Σ = (1, 2, 3, ... , n) is a sequence of elements in the alphabet where every element occurs precisely once. Sn denotes the set of all such permutations on a given alphabet. In ∈ Sn be the Identity permutation where elements are in ascending order i.e. (1, 2, 3, ... , n). Rn ∈ Sn is the reverse permutation where elements are in descending order, i.e. Rn = (n, n − 1, n − 2, ... , 2, 1). An operation has been defined in OEIS which consists of exactly two moves: set-rotate that we call Rotate and pair-exchange that we call Exchange. Rotate is a left rotate of all elements (moves leftmost element to the right end) and Exchange is the pair-wise exchange of the two leftmost elements. We call this operation as LE. The optimum number of moves for transforming Rn into In with LE operation are known for n ≤ 10; as listed in OEIS with identity A048200. The contributions of this article are: (a) a novel upper bound for the number of moves required to sort Rn with LE has been derived; (b) the optimum number of moves to sort the next larger Rn i.e. R11 has been computed. Sorting permutations with various operations has applications in genomics and computerinterconnection networks.