An Upper Bound for Sorting $R_n$ with LRE

Abstract: A permutation $\pi$ over alphabet $\Sigma = {1,2,3,\ldots,n}$, is a sequence where every element $x$ in $\Sigma$ occurs exactly once. $S_n$ is the symmetric group consisting of all permutations of length $n$ defined over $\Sigma$. $I_n$ = $(1, 2, 3,\ldots, n)$ and $R_n =(n, n-1, n-2,\ldots, 2, 1)$ 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, right-rotation 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 $R_n$ into $I_n$ with $LRE$ operation are known for $n \leq 11$ 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 $S_n$ when generated by $LRE$ operations \cite{oeis}. The contributions of this article are: (a) The first non-trivial upper bound for the number of moves required to sort $R_n$ with $LRE$; (b) a tighter upper bound for the number of moves required to sort $R_n$ with $LRE$; and (c) the minimum number of moves required to sort $R_{10}$ and $R_{11}$ 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.