Perfect sequence covering arrays

An (n, k)-perfect sequence covering array with multiplicity is a subset of the n! permutations of the sequence (1, 2, · · · , n) whose elements collectively contain each ordered length k subsequence exactly times. The primary objective is to determine, for given n and k, the smallest value of (denoted g(n, k)) for which such a configuration exists. In 2020, Yuster determined the first known value of g(n, k) greater than 1, namely g(5, 3) = 2, and suggested that finding other such values would be challenging. We determine that g(6, 3) = g(7, 3) = g(7, 4) = 2 and g(8, 3) 2 {2, 3} by modifying an old search algorithm due to Mathon and van Trung, and by restricting a perfect sequence covering array to be a union of cosets of a prescribed nontrivial subgroup of the symmetric group Sn. This prescribed structure provides a deeper understanding of the existence pattern for perfect sequence covering arrays by combining combinatorial and algebraic viewpoints.