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.

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1 Abstract-A permutation π over alphabet Σ = 1, 2, 3, . . . , n, is a sequence where every element x in Σ occurs exactly once. Sn is the symmetric group consisting of all permutations of length n defined over Σ. In = (1, 2, 3, . . . , n) and Rn = (n, n − 1, n − 2, . . . , 2, 1) are identity (i.e. sorted) and reverse permutations respectively. An operation, that we call as an LRE operation, has been defined in OEIS with identity A186752. This operation is constituted by three generators: left-rotation, rightrotation and transposition(1,2). We call transposition(1,2) that swaps the two leftmost elements as Exchange. The minimum number of moves required to transform Rn into In with LRE operation are known for n ≤ 11 as listed in OEIS with sequence number A186752. For this problem no upper bound is known. OEIS sequence A186783 gives the conjectured diameter of the symmetric group Sn when generated by LRE operations [1]. The contributions of this article are: (a) The first non-trivial upper bound for the number of moves required to sort Rn with LRE; (b) a tighter upper bound for the number of moves required to sort Rn with LRE; and (c) the minimum number of moves required to sort R10 and R11 have been computed. Here we are computing an upper bound of the diameter of Cayley graph generated by LRE operation. Cayley graphs are employed in computer interconnection networks to model efficient parallel architectures. The diameter of the network corresponds to the maximum delay in the network.

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