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In this paper, we explore the more complex $n = 2m-4$ case, with several forms of additional permutations. We characterize $s^{m-4}(S_{2m-4})$ and find that its size is \\[B_m + \\frac{m^2 + 7m - 28}{2}\\] for $m \\geq 5$. This answers Defant's question about the $2m-4$ case. 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