Efficient Enumeration of Graceful Permutations

A graceful \( n \)-permutation is a graceful labeling of an \( n \)-vertex path \( P_n \). In this paper, we improve the asymptotic lower bound on the number of such permutations from \( \Omega\left(\left(\frac{5}{3}\right)^n\right) \) to \( \Omega\left(2.37^n\right) \). This is a computer-assisted proof based on an effective algorithm that enumerates graceful \( n \)-permutations. Our algorithm is also presented in detail.